New York is 1-1 on its eight-game road trip while the Dodgers had a four-game winning streak snapped.
Belisario (4-6) issued a leadoff walk in the ninth, who was immediately erased on a fielder's choice grounder off the bat of Robinson Cano.
Cano moved to second on a ground out by Alfonso Soriano, after which Ichiro Suzuki was intentionally walked. Overbay followed with a single to center that brought home Cano and the Dodgers then suffered a miscue that allowed the Yankees to add two runs.
Both Suzuki and Overbay scored when a grounder by Jayson Nix was botched for an error by second baseman A.J. Ellis.