Suzuki's homer in the sixth inning made it 3-0 Yankees and his two-run single in the seventh capped the scoring for New York, which triumphed over Los Angeles in the Dodgers' first-ever regular-season game at Yankee Stadium. Suzuki finished 3-for-4.
Lyle Overbay also drove in two runs for New York, which last hosted Los Angeles in Game 6 of 1981 World Series.
Hiroki Kuroda (7-5) earned the victory with 6 2/3 innings of one-run ball, allowing eight hits and a walk while striking out two.
Mariano Rivera finished up with his 25th save as the Yankees took the rain-delayed opener of a three-game series.
Dodgers starter Hyun-Jin Ryu (6-3) was tagged with the loss after giving up three runs on five hits in six frames.
Hanley Ramirez went 4-for-4 with a two-run homer for Los Angeles.