Bruce's blast off Mets starter Shaun Marcum with none out in the sixth snapped a 3-3 tie and sent Cincinnati on its way to an eighth win in 10 contests.
Bruce also supplied an RBI double and Brandon Phillips drove in a pair of runs as the Reds claimed the opener of a three-game series.
Cueto (2-0), making his first start since April 13 after missing time with a strained lateral muscle, earned the victory, yielding three runs on three hits over five full frames.
Marcum (0-5) surrendered four runs on six hits over six full frames.
Marlon Byrd smacked a three-run homer for the Mets.
Celebrity Families of 2014 [PHOTOS]