The AL Central-leading Tigers were trying for their best string since 1934, when they tied the team record of 14 straight wins. That run ended across the street at the old Yankee Stadium when Babe Ruth and Lou Gehrig each drove in runs to down Detroit.
Gardner singled just past Cabrera's dive at third base. Al Alburquerque (2-2) took the loss and Shawn Kelley (3-1) got the win.
Austin Jackson got three doubles and four hits overall for Detroit. His one-out double began Detroit's comeback in the ninth.
With some scattered fans wearing No. 13 jerseys, Rodriguez finished 0 for 4 in his first game at Yankee Stadium since last October. The sellout crowd of 46,545 let him have it the whole night, from the moment the lineups were announced to the time he looked at a slider for his third strikeout of the game.
A-Rod made his season debut Monday in Chicago against the White Sox, hours after Major League Baseball handed him a 211-game drug suspension. The three-time AL MVP can play until there is a decision on his appeal, likely after the season.
Jayson Nix replaced Rodriguez in the top of the ninth.
Nix walked to start the 10th and scored the winning run.
Cabrera delivered a most dramatic hit an inning earlier.
After a foul ball left him hopping, Cabrera got a checkup from manager Jim Leyland. Fact is, Cabrera almost didn't make it that far moments before that, first baseman Lyle Overbay reached over a railing and barely missed catching Cabrera's foul pop that would've ended the game.
But Cabrera regrouped and lined a 2-2 pitch over the wall in dead center field, slowly making his way around the bases and silencing the crowd. He drove in Detroit's first three runs, and now has 34 homers and 108 RBIs.
For Rivera, it was his second straight failure the first time that happened since 2011 for the career saves leader. Rivera has blown four in 39 chances this year, including a ninth-inning miss Wednesday night in Chicago against the White Sox.
Yankees starter Ivan Nova allowed one run in seven innings. He wiggled out of a jam in the first, striking out Victor Martinez and Andy Dirks to leave the bases loaded.
Nova cruised until Jackson led off the seventh with a double. Torii Hunter was then called out on strikes and objected to the call by Triple-A fill-in umpire Will Little.
After Cabrera followed with an RBI single, Little ejected Tigers pitcher Max Scherzer for hollering from the dugout.