Trailing 3-2 starting their final at-bat, the Mets got a man aboard with one out when Ruben Tejada was hit by a pitch delivered by Marlins reliever Steve Cishek (0-1).
Kirk Nieuwenhuis followed with a single, sending Tejada to third, while Nieuwenhuis gained second when outfielder Juan Pierre threw to third.
That brought up Byrd, who lined a hit past drawn-in third baseman Chris Valaika, scoring both runners for the victory as the Mets took the rubber match of their three-game set.
Scott Rice (1-0) earned the win with a scoreless top of the ninth, overcoming a pair of walks.
Donovan Solano went 3-for-5 with an RBI while Valaika and Justin Ruggiano each provided two hits and a run batted in for Miami.
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