The Reds trailed 2-0 for much of the game but turned it around in the sixth when Frazier's run-scoring single triggered a three-run outburst against Dodgers starter Chad Billingsley (4-8).
Mesoraco added a two-run double in the inning and Cincinnati went on to score five more times over the final two frames, keyed Frazier's RBI triple and another run-scoring hit from the Cincinnati catcher.
Homer Bailey (6-6) hurled eight innings to garner the victory for the National League Central-leading Reds, allowing a pair of runs on five hits while walking one and striking out seven.
Billingsley surrendered three runs on eight hits over six full frames for the second-place Dodgers, who fell 1 1/2 games behind idle San Francisco in the NL West with their 10th loss in 12 games.
Jerry Hairston Jr. socked a solo homer for Los Angeles.