Headley's blast in the first inning put the Padres ahead to stay, while Alonso's clout in the third handed San Diego a 5-0 lead.
Quentin also contributed a run-scoring double for the Padres, who took the rubber match of a three-game series.
Jason Marquis (5-6) collected the victory, yielding three runs on four hits over 6 1/3 frames. The right-hander won for the fourth time in five starts, allowing one walk while striking out five.
New York's Matt Harvey (1-2) absorbed the loss in his third major league start, surrendering five runs on eight hits over five full frames.
Ronny Cedeno cracked a pinch-hit, two-run homer for the Mets.
Swim Week Miami Beach 2014 [PHOTOS]