Polanco singled with one out in the 20th against Shaun Marcum (0-7) and then came home on consecutive singles by Rob Brantly and Hechavarria as the Marlins scored their first run since the fourth.
Kevin Slowey (2-5) pitched seven shutout innings of relief to pick up the win and Steve Cishek fired a 1-2-3 bottom of the 20th for Florida, which on April 30 outlasted the Mets in a 15-inning affair.
Coupled with the Toronto Blue Jays' 18-inning win over Texas earlier Saturday, it marked just the second time in MLB history two games had gone at least 18 innings on the same day.
Marcum had hurled seven shutout innings for New York prior to the 20th, retiring 15 straight batters at one point.
Juan Lagares got the only RBI for the Mets, who dropped the opener of a rain-shortened two-game series.
2014: The Year in Music [PHOTOS]