Alex Rodriguez, playing at home for the first time since admitting he used steroids and having hip surgery, was 0 for 1 with a strikeout and four walks.
Brett Gardner, who entered after Johnny Damon was ejected in the third, began the comeback in the seventh with the first inside-the-park homer at the $1.5 billion ballpark.
Gardner, who went 3 for 3, tripled leading off the ninth against Nathan (1-1) and scored on a single by Mark Teixeira to cut the deficit to 4-3.
Rodriguez walked, Hideki Matsui struck out and Nick Swisher hit a bouncer between first and second that Morneau snagged with a dive and turned into an out at first.
Robinson Cano then was intentionally walked to load the bases, and Cabrera hit the first pitch into short left-center to drive in two runs.
Jose Veras (2-1), New York's sixth pitcher, struck out his only batter for the win.
Rodriguez was cheered before he walked in the second, then was booed when he struck out with the bases loaded and one out in the third.
Five more homers were hit. There have been 52 homers at Yankee Stadium, two shy of the major league record for most in the first 14 games at a major league park. Thirty-five homers have been to right field.
And there was some testiness when Teixeira yelled and pointed at Carlos Gomez, who ran past him on the fair side of the foul line on his eighth-inning sacrifice. Players converged and screamed, but no punches were thrown.
The Yankees hadn't take the lead in 48 consecutive innings in their own ballpark before the ninth-inning rally.
Morneau homered in the second and fifth against Phil Hughes for his 10th multihomer game and first since July 23, 2007. Mauer homered in the seventh off Phil Coke to back Francisco Liriano, who allowed one run, four hits and six walks in six innings.
Derek Jeter homered for New York, and Gardner circled the bases in the seventh on a looper down the left-field line that took a topspin bounce away from Denard Span and went to the wall. Gardner slid home headfirst easily ahead of the throw.
