Joba Chamberlain (5-2) picked up the win, allowing a run and three hits with eight strikeouts in 6 2/3 innings.
The contest, which completed a three-game sweep, marked a celebration of the Yankees' 63rd Old-timers Day.
The Yankees had not taken a series from the Tigers since May of 2005.
Rodriguez's homer, his 572nd, moved him one behind Harmon Killebrew for ninth place on the career list.
Clete Thomas homered for the Tigers.