In the musical 1776, Congress needs a unanimous vote to declare independence. Right when John Dickinson is about to tell the Congress that Pennsylvania votes nay, Benjamin Franklin gets his Chessmaster on and asks that the three delegates be polled. Dickinson and Franklin cancel each other out by voting "nay" and "yea" respectively, leaving everything up to Judge James Wilson, who's spent the entire production mindlessly seconding everything Dickinson says. But the entire question of American independence ends up in his lap, when, as Franklin tells him, "every mapmaker in the world is waiting for your answer," he realizes he doesn't want to be known forever as the man who stopped American independence.
The entire sequence of "The Deal (No Deal)" from Chess. Molokov wants the USSR to win the chess tournament. De Courcey wants American prisoners in the USSR to be released. They work together to get defected Soviet champion Anatoly to throw the match, by A) pressuring his abandoned wife Svetlana to pressure him lest they make life difficult for her and her children, as well as B) pressuring his new lover Florence to pressure him by offering her the chance to see her Disappeared Dad again. They also bring in Unwitting Pawn Freddie to try and persuade Anatoly to throw the match for Florence's sake, but after a crushing rejection from both of them (and a show-stopping musical number) he turns on Molokov, De Courcey, and indeed everyone, and tells Anatoly how and why to win.
Results vary in each version of Chess, though Anatoly ends up going back to the Soviet Union in all of them. In the London version, he wins the game; in others, he loses.
In The Phantom of the Opera, Raoul and the others pressure a fearful Christine into performing the lead role in the opera that the Phantom has written. Knowing full well that the Phantom will attend if she performs, they plan to catch him then. While this might not be what the Phantom wants precisely, he has anticipated it, and as such, is able to disguise himself and abduct Christine.
In Sherlock Holmes, Holmes risks his life to negotiate the purchase of a MacGuffin from the villains, not letting Alice know he knows it's a fake in order to manipulate her into surrendering the real MacGuffin to the Count and Sir Edward, who congratulate Holmes for pulling off this ingenious scheme.
In its original form, and shorn of its set pieces, The Nutcracker is a Batman Gambit by Herr Drosselmeyer, who sets up his goddaughter Clara to help him free his nephew, the Prince.
Twelfth Night has two of these, both of which go entirely according to plan. First, Maria plants a fake love letter to Malvolio from Olivia, instructing him to do ridiculous things in order to win her heart; this only works because Maria knows Malvolio is vain and self-deluding enough to eat it up. Later, Sir Toby has a little fun by pitting two known cowards against each other in a fight and then informing each duelist of his respective opponent's berserker tendencies. Incidentally, Sir Toby and Maria end up eloping by the end of the play, which just goes to show you that this trope can be a highly effective form of romantic bonding.