Alexander the Great repeatedly outgambitted Darius during his final advance on the Persian Empire. Darius employed scorched earth on the road Alexander was most likely to take, but Alexander foresaw it and took another; Darius had the plains of Gaugamela flattened in order to better use his war chariots on Alexander's army, but Alexander started moving his army out of the flattened section to force his hand and then countered the chariots with his own tactics; and at the end, in midst of the great pitched battle Darius had sought in order to crush Alexander with his much larger armies, Alexander revealed to have his own plan and exploited a breach in the Persian army to thrust his Companion cavalry against Darius himself, who lost his nerve (who could blame him by this point, though) and just ran away to live another day.
The Battle of Ilipa during the Punic Wars was a series of outgambits by Publius Cornelius Scipio against Hasdrubal Gisgo and Mago Barca. Right from the start, the Punics attempted to attack with cavalry before the Romans could finish camping, but it turned out Scipio had foreseen this and placed his own cavalry hidden, catching their opponents on the flank and routing them. Scipio then passed the next days forming his army outside and withdrawing back and forth than engaging the Carthaginians in pitched battle, not without letting them see his choice of formation every time, with his heavy Roman legions in the center and his lighter Hispanic allies in the wings. Hasdrubal and Mago deployed their troops to counter this, with their Libyan heavy infantry on the center and their own Hispanic vassals and mercenaries on the wings, knowing that their significantly larger army would eventually unbalance the Mirror Match and prevail by attrition. However, the day Scipio actually attacked, he came out with his formation reversed, with the legions on the wings and the Hispanics on the center, and the shocked Punics couldn't rearrange their ranks because the Roman Hispanics were specifically there to stop it. The legionaries destroyed the weaker Punic wings, and eventually the Carthaginian center crumbled and routed under pressure from the sides.
After Ilipa, Scipio courted King Syphax of Numidia into allying with Rome against Carthage, and after succeeding, he sent military trainers in order to turn the disorganized Numidian warriors into a force capable to resist the Punics. However, this time Hasdrubal Gisgo countered by literally courting Syphax by giving him his daughter Sophonisba, who completely charmed the Numidian king into changing sides, thus turning the Roman local efforts to Carthage's favor. Now, this would cause Syphax's side to be outgambitted in turn when the king, over-ambitious with his Roman-trained army and his friendship with Carthage, became eagerly open to diplomacy with Scipio in an attempt to regain Rome's favor too (even offering to mediate between Carthage and Rome). Scipio only capitalized on this to find the weak points of the enemy camps through spywork and lull the Africans into a false security by pretending to consider their deal, and when nobody expected it, he ordered a sudden fire attack on the camps, destroying them and wiping out Syphax' and Hasdrubal's armies.
Why Roman emperor Tiberius remained in power 'till his natural death in spite of being widely hated by the Senate and surrounded by untrustworthy allies: he could easily play them against each other without even leaving his villa on the island of Capri. Best shown by the fall of Sejanus, the chief of the Praetorian Guard who was using his role as Tiberius' representative in Rome to prepare for a coup: upon learning of his treachery, Tiberius started sending out letters and orders made specifically to confuse the situation and undermine his authority in general and over the Praetorians in particular while at the same time showing the utmost respect for Sejanus, then, once the groundwork was laid, he sent to Rome a man named Macro and had him summon the Senate and Sejanus for the public appointment of Sejanus to tribune (thus effectively making him as powerful as Tiberius himself)... Except the letter, before announcing the appointment, suddenly ordered the arrest of Sejanus and two senators loyal to him, and while the letter was being read the Praetorians guarding the place had been replaced by vigiles (Rome's police force and firefighters) and Macro had announced the Praetorians that Tiberius had just appointed him to their command.
Early Modernity
During their their conquest of America, the Spaniards and Portuguese found that an uncanny lot of hostile indigenous tribes had the tactic of pretending to be friendly, luring the bearded foreigners into their villages or cities, and then attack and kill everybody (at least two important names, Juan Díaz de Solís and the eponymous son of Diego García de Paredes, were killed this way), so the Iberians eventually wised up and started attacking first whenever they felt they were guided into an ambush. Of course, there's space to speculate that some of those traps might have been less real than imagined, caused variously by genuine paranoia, greedy captains putting up excuses to sack villages, or manipulation by indigenous guides who happened to hail from rival tribes ("they are totally going to betray and kill us, capitán, we should strike first - and on that note, can I keep their chief's wives when it's over?").
During their journey to the capital of the Aztec Empire, Hernán Cortés and his various allies crossed through the territory of Cholula, a state vassal to the Aztecs that Cortés wanted to visit in order to eye the empire's forces. Upon arriving in Cholula, however, they were met by fishy chieftains and a city suspiciously low on civilians, and to their greater concern, Cortés' indigenous allies found fortifications in the streets and around the city. Cortés had some local noblemen interrogated, eventually finding out that the Aztecs had ordered them to ambush and kill the foreigners, and decided to turn the trap on its head. Next morning, he asked the Cholultecs to form up their people in militay age outside, after which he confronted them with his army, accused them of traitors, and ordered his men to attack, causing the subsequently named Massacre of Cholula. After the dust had set, Cortés sent a messenger to the Aztec capital to keep diplomacy, claiming he had just trounced a wicked conspiracy the Aztecs were surely not part of, and installed the terrified noblemen back in their seats with the warning to never try anything of the like again (they didn't).
A much less successful example happened when Cortés had to march against Pánfilo de Narváez and left his lieutenant Pedro de Alvarado in command of Tenochtitlan. Apparently, when the Aztecs asked for permission to perform one of his festivals, Alvarado was informed by indigenous informants that the whole thing was a trap to revolt against him. Alvarado, who in commanding matters was largely a Dumb Muscle, didn't come up with anything better than to try to pull a Cholula on the capital of the Aztec Empire, allowing them to host the darn festival only for him to attack them and butcher the unarmed participants in front of everybody. It's unknown whether the thing was really a trap (there are hints that it wasn't, and that the native allies only wanted to crash the festival because it was going to feature human sacrifice of prisoners of their own nation), but if it wasn't, it became a Self-Fulfilling Prophecy and caused the total revolt of the Aztec Empire.
Francisco Pizarro and his people were marching to meet the Inca Emperor Atahualpa in Cajamarca when one of his native spies, the Tallán chieftain Guachapuro, returned with the news that Atahualpa had evacuated Cajamarca and surrounded it with giant armies, all but stating he planned to ambush the Spaniards. Apparently, Atahualpa had planned to capture and enslave the Iberians as artisans in order to get their advanced weapons and crafts (he was even under the impression that the Spaniards had an art to make people immortal, as an Inca spy had seen a Spanish barber at work and believed he had the power to rejuvenate people). When they arrived, Pizarro continued with his plans and exchanged seemingly friendly embassies with the emperor, but when an overconfident Atahualpa visited the Spanish camp in one of them, Pizarro seized the chance and ordered to attack. The Inca were routed in midst of the confusion, their emperor was captured, and when the news of the event reached the corners of the empire, many tribes and states previously unhappy with Atahualpa came to congratulate Pizarro, gifting him with support and armies for the rest of the conquest.
20th century
A well-known military example is the Battle of Midway during World War II. By mid-1942, Japan was seeking to lure out the American carriers, which at that point were the only major threat to Japanese naval domination in the western Pacific. Japanese Admiral Yamamoto figured that attacking Pearl Harbor a second time was now too risky, because land-based aircraft from the US mainland had been transferred in. Hence, an attack on Midway Island was decided, as Yamamoto figured the US could not afford to let that island fall to Japan, but it was out of range of land-based aircraft. This in turn would end up forcing the US to send their carriers out, where they would be targeted by Japan's own carriers. However, unbeknownst to Japan, the US Navy had already broken the Imperial Japanese Navy's communication codes and learned of the plan, allowing them to craft a counter-trap at Midway. The battle which followed resulted in the IJN losing four carriers while the US Navy lost only one. The US could could afford to lose one carrier, because they could eventually replace the loss. But Japan simply couldn't replace four lost carriers, not having anywhere near the industry or manpower to match America's production.
