Arroyo (2-1) limited the Brewers to one run on six hits over 6 2/3 innings, issuing only one walk while striking out nine.
The Reds backed him with five runs in the fourth, during which Bruce and Zack Cozart hit home runs and Joey Votto contributed a run-scoring double.
Miguel Cairo added an RBI for Cincinnati, which won for the fourth time in five games and took the opener of a three-game set at Miller Field.
Marco Estrada (0-2) absorbed the loss for Milwaukee, surrendering five runs on eight hits in a seven-inning start.
Ryan Braun went 2-for-4 with a solo home run as the Brewers fell to 1-5 in May.
Dennis Rodman pledges to end trips to North Korea
Jessica Simpson shares three-way kiss with friends in photo