Lidge (0-2) came on in the ninth and immediately got in trouble.
Johnny Damon drew a leadoff walk and stole second. After Mark Teixeira struck out, Rodriguez smacked a full-count pitch over the short porch in right for his seventh homer - and 10th hit - after missing the first 28 games this season following hip surgery.
Robinson Cano followed with a single, stole second and scored without a play on Cabrera's single to right-center.
It was the third game-ending hit of the season for Cabrera, who raised his arms as he rounded first and dashed with glee all the way across the diamond as his jubilant teammates chased him down to celebrate.
The Yankees have won 10 of 11 and are making a habit of these walk-off victories. They had three straight against Minnesota last weekend, and capped this one in the same way: pitcher A.J. Burnett pelted Cano and Cabrera with a pie in the face as they waited to do an on-field interview - even getting some whipped cream on radio broadcaster Suzyn Waldman.
For Rodriguez, it was his sixth homer in the past eight games.
Lidge, who converted all 48 save chances in 2008 - including the postseason - during his first year in Philadelphia, has blown three of 11 opportunities this season.
Jose Veras (3-1) got one out in the eighth for the win.
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