After throwing 31 pitches in the first inning, when the Royals managed their only run off him, Pettitte needed just 61 pitches through the next 5 2-3 innings.
Cabrera tied the game at 1 with a solo homer in the third, driving a 1-2 pitch to left off John Bale, the first left-hander the Yankees have faced this year.
The Yankees seized a 2-1 lead in the fourth when Jorge Posada, who led off the inning with a single and advanced to third on Hideki Matsui's base hit, scored when Jose Molina grounded into a double play.
Cabrera and Rodriguez each had an RBI in the fifth. Cabrera's single scored Morgan Ensberg, and Johnny Damon scored on Rodriguez's fielder's choice ground out after his own infield single.
Rodriguez and Posada added back-to-back home runs in the ninth off Hideo Nomo, who replaced Bale for his first appearance in the majors since July 15, 2005, while with Tampa Bay.
Nomo, who is making a comeback at 39 and after having elbow surgery in 2006, worked out of a bases-loaded jam in the seventh. He then got through the eighth without any damage before giving up the homers to Rodriguez and Posada with two out in the ninth.
Rodriguez increased his career RBI total to 1,510, moving him one ahead of Mickey Mantle and into 45th place on the all-time list.
Bale (0-2) left after six innings, allowing four runs and eight hits with a walk while striking out three.
Jose Guillen's two-out double in the first scored Mark Teahen for the Royals' only run.
The Yankees, who were 1-for-13 with runners in scoring position in the first two games of the series, went 2-for-8 on Thursday night. They are 13-for-74 with runners in scoring position in the first 10 games of the season.
