Monroe was 2-for-5, Chris Shelton went 2-for-3 with an RBI, and Brandon Inge doubled and drove in a run for Detroit, which has won four straight.
The Devil Rays have lost ten in a row.
In relief of starter Nate Robertson, Franklyn German improved to 3-0.
Robertson was ejected after throwing one pitch. Home plate umpire Tim McClellan immediately tossed Robertson after his behind-the-back pitch.
Nick Green hit a two-run homer for Tampa Bay. Loser Scott Kazmir sank to 3-7, allowing five runs and five hits in just three innings with four walks and two strikeouts.