Stephen Drew, Willie Bloomquist and Aaron Hill all reached base to lead off the game against Bronson Arroyo (4-6) and Montero followed by slicing a double into the left-field corner to chase home all three.
The Arizona catcher then scored on Justin Upton's single and added a solo homer in the third for the D-backs, who were coming off a three-game sweep at the hands of the Chicago Cubs.
Upton finished 3-for-4 while Bloomquist chipped in with three hits for the Diamondbacks, who took the opener of a four-game series.
Wade Miley (10-5) allowed three runs over the first two innings but then settled down, eventually surrendering eight hits and striking out six over 5 2/3 innings.
Arroyo was tagged for five runs on eight hits over three innings for the Reds, who came into the contest with a one-game lead over Pittsburgh in the National League Central.
Drew Stubbs drove in a pair of runs for Cincinnati.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys