The Yankees tied it at 1 on Vernon Wells' RBI single in the fourth, but Price avoided further damage by getting a fly ball from Ben Francisco and Lyle Overbay's double-play grounder.
Ichiro Suzuki's double-play grounder put the Yankees ahead and they held on to take two of three games in a series between the teams with the highest and lowest payrolls in the majors.
The Yankees tied it at 1 on Wells' RBI single in the fourth, but Price avoided further damage by getting a fly ball from Ben Francisco and Overbay's double-play grounder with runners on first and second with no outs.
Kuroda's eighth inning ended with a snappy doubleplay started by shortstop Jayson Nix, who ranged far to his left to scoop up Nate McLouth's grounder up the middle and flipped to Robinson Cano covering the bag for a 6-4-3 doubleplay.
Robinson's hard slide into second broke up a potential doubleplay and allowed a run to score in a two-run seventh inning that put the Canadians ahead 7-3.
Stoudemire's highly efficient play of late seems almost certain to draw occasional double teams from opposing defenses as they study the Knicks on film going forward.
Wolves chief executive Andy Gatcliffe welcomed the double signing after a difficult weekend, which saw Warrington's play-off hopes ended with a 46-8 defeat at Catalans Dragons.
Munenori Kawasaki reached on an infield single, went to third on Cabrera's base hit and scored on a throwing error by Eduardo Nunez as the Yankees failed to turn a doubleplay.