Quintanilla's two-run single highlighted the big inning, during which the Mets broke open a 1-1 tie by sending 11 men the plate and banging out seven hits.
Anthony Recker supplied a solo homer and two RBI while Eric Young also drove in two runs for New York, which has taken the first two contests of their four-game home series.
Jeremy Hefner (3-6) tossed seven innings of one-run ball to earn the victory, limiting the Diamondbacks to four hits and two walks while striking out six.
D-backs starter Patrick Corbin (9-1) absorbed his first loss of the season in his 17th start. The left-hander was tagged for five runs on six hits over six full frames.
Martin Prado went 3-for-4 with a solo homer for National League West-leading Arizona, which dropped a fifth straight and eighth in nine games.