Montero, the designated hitter who was playing in only his fourth major league game, launched a solo shot off Jim Johnson (5-5) in the fifth inning to put the Yanks ahead for good at 9-8 and added a two-run shot in the seventh.
Cano's slam came during a six-run second inning in which Mark Teixeira also went deep for the Yankees, who came into the game with a 1 1/2-game lead over Boston atop the American League East.
Aaron Laffey (2-1) earned the victory after getting the final out of the fifth and the first two outs of the sixth for New York.
Luis Ayala hurled 1 2/3 frames of scoreless relief and Mariano Rivera collected his 38th save in the win.
Mark Reynolds clubbed a two-run homer while Nick Markakis and Robert Andino added solo shots for the Orioles, who dropped the opener of a three-game series at Yankee Stadium.