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.
Pistorius testifies he didn't consciously pull trigger when he shot girlfriend
Pot vending machine to debut