With the score tied 4-4, Morrison led off the 10th with a single off of Bobby Parnell (4-3), went to second on a Justin Ruggiano hit and came home when Mets second baseman Daniel Murphy mishandled Adeiny Hechavarria's potential double-play grounder.
After Robert Carson came in to pitch, Greg Dobbs drove in Ruggiano a sacrifice fly and Olivo followed with a two-run shot as the Marlins outhit the Mets 17-6 and swept a rain-shortened two-game series.
The uprising made a winner of Chad Qualls (1-0), who recorded the final two outs in the bottom of the ninth.
Murphy went 2-for-3 with a solo homer and two RBI for Mets, who tumbled to a sixth loss in seven games.
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