Sabathia bounced back from an erratic opener, staying away from too much trouble against Josh Hamilton and Texas' big bats. Rangers slugger Nelson Cruz made an early exit with hamstring trouble, a day after Yankees star Mark Teixeira was lost for the postseason with a hamstring injury.
Now the teams will return to Texas to decide the best-of-seven series. When they resume Friday night, Phil Hughes starts for the defending World Series champion Yankees against Colby Lewis in a rematch of Game 2, won by Texas 7-2.
Facing a 3-1 deficit in the best-of-seven series, the Yankees showed some spark at the plate against C.J. Wilson.
Jorge Posada and Curtis Granderson had back-to-back RBI singles in the second inning after Wilson walked two. Granderson's hit to right field set off a series of bad throws that led to two more runs on the play and left Derek Jeter smiling in the New York dugout.
Swisher led off the third with a long drive to left. Cano, batting in the No. 3 spot in place of injured slugger Mark Teixeira, followed with his fourth homer of the series, giving New York, which entered batting .198 for the ALCS, a 5-0 lead under a silvery red twilight sky in the Bronx.
Wilson held the Yankees scoreless over the first six innings in his Game 1 matchup against Sabathia before Cano connected in the seventh and started the Yankees' big comeback for their lone win in the series.
Wilson was done after five ineffective innings Wednesday. He yielded six hits and six runs five earned and walked four, one intentionally.
Lance Berkman took over at first base for Teixeira, and he gave the Yankees a scare in the fourth inning when he wiped out while chasing a foul popup and landed hard on his back. Berkman rolled over onto his stomach as the Yankees came out to check on him, and the hushed crowd waited for several tense moments before he got to his feet and remained in the game.
The switch-hitter had a bases-loaded sacrifice fly in the fifth inning to make it 6-1.
Texas slugger Nelson Cruz was replaced by David Murphy in left field to start the fifth because of a tight left hamstring. He is day to day.
Pitching on regular rest for the first time this postseason, Sabathia labored through his third October start. He allowed 11 hits and failed to record a perfect inning.
Matt Treanor homered to start the fifth and had an RBI groundout in the sixth, when the Rangers loaded the bases. Sabathia got out of this jam by striking out Mitch Moreland looking. The big lefty gave a hearty shout and double-fist pump by his waist after the breaking pitch froze the first baseman.
He wriggled out of two other jams with help from a pair of double plays and six more strikeouts. Sabathia did not walk a batter in throwing 112 pitches.
Elvis Andrus singled leading off the afternoon and has at least a hit in all 10 postseason games. But Ian Kinsler bounced into a double play. In the fifth, he got hot-hitting Josh Hamilton, who had a three-run homer off Sabathia in the first inning of Game 1, to ground into an inning-ending 4-6-3 double play with runners on first and second.
Texas had outscored New York 18-3 in the first two games at Yankee Stadium. The Yankees have not lost three straight home games in a single series in the postseason since the 1942 World Series.
After Alex Rodriguez and Lance Berkman walked in the second, Posada singled in A-Rod for a 1-0 lead. Granderson then hit a liner to right field that scored Berkman and Jeff Francoeur tried to nail Posada advancing to third but his strong throw bounced passed third baseman Michael Young for an error. Wilson raced over to pick up the ball in front of the Rangers' dugout and his throw sailed over catcher Matt Treanor's head and Posada touched home safely for a 3-0 lead.
Sabathia gave up five runs and six hits over four innings in Game 1 of the ALCS, pitching after an eight-day layoff.