Well, the person in question doesn't actually HAVE to be saved to be this trope, the people involved just have to BELIEVE it. The quote still fits so long as people THOUGHT saving her would save the world, so I guess it depends on that...
I think Eureka from the Eureka Seven anime would qualify. At the end of the series, Gekkostate has to save Eureka from becoming the replacement control center for the Skub Coral, if they fail planetary genocide occurs.
From the archives:
Name suggestions for giving this one a better name include "Save the Cheerleader" and "Saving the Cheerleader."
The Butterfly of Doom page would benefit as well: "When the nail in For Want Of A Nail is a person instead of a thing or object, that's Saving The Cheerleader."
This was about to be put to a vote, but then the discussion got archived and forgotten when the new system was implemented.
Now comes the question: is anyone still interested in putting this to a vote?
The article page already agrees that "Save This Person, Save the World" is in need of a better name.
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I think simply changing the "This Person" to "the Cheerleader" would work. Although Heroes is nowhere near as popular as it was, even people who don't know anything about Heroes know the line "Save the Cheerleader Save the World".
"Men are still good. We fight, we kill, we betray one another, but we can rebuild. We can do better. We will. We have to." - Bruce Wayne
And I guess it also has logic in that the characters who play the roles of The Cheerleader are usually the type who cause the most heroic angst when they die.
I think the "Saving" at the start is a must, myself. It doesn't HAVE to be a cheerleader type character who has to be protected, it doesn't even have to be someone the heroes like. They just have to not die.
Dropping this quote. It looks very much like a straight example of the trope, but it's actually a subversion: the show invests a lot of time and effort into setting up the trope but then the cheerleader in question isn't saved and they save the world anyway. Much less misleading for a quote to be a straight example, especially the page quote.