Cole Gillespie broke his bat grounding out in the 13th with runners on second and third, and Gregor Blanco grounded out with the bases loaded in the 15th.
Posey's 13th homer put the Giants ahead 2-0 in the first.
Byrd singled leading off the fifth for the first hit off Lincecum, and John Buck's double sliced San Francisco's lead to 2-1.
Young tripled leading off the sixth, and Murphy followed with the tying single. Murphy advanced to third on Scutaro's throwing error at second base and scored when Ike Davis grounded into a double play to give the Mets a 3-2 lead.
Lincecum had a season-high 11 strikeouts in seven innings. He gave up three runs, six hits and one walk. It was the 33rd time Lincecum has had at least 10 strikeouts in a game and the first time this year.
But the two-time NL Cy Young Award winner was nowhere to be found in the Giants' clubhouse after the game. A team spokesman said he had already left to recover — as did many of his teammates.
Other than Posey, Harvey had his way with San Francisco's lineup until late. He allowed three runs and six hits, striking out six and walking one.
Pence, who made a pair of diving catches in right field, tripled leading off the seventh. The ball hit the cement near the top of the right-field wall and ricocheted back onto the field.
Umpires reviewed the play and ruled that the triple stood, though the decision turned out to be an afterthought. Crawford immediately singled home Pence for the tying run.
"There's frustration," Pence said. "We want to stay as positive as we can right now. We've got to get it done. We've got to keep pushing to find a way to turn it around."
NOTES: Posey will get Tuesday night's game off after catching 16 innings, Bochy said. ... Mets LHP Jonathon Niese (shoulder inflammation) has been cleared to start throwing again. The team expects him back at some point this season. ... Mets RHP Shaun Marcum had his right hand examined by a specialist in St. Louis. With an off day Thursday, Harvey is slated to take Marcum's spot in the rotation Saturday at Pittsburgh. ... Barry Zito (4-6, 4.44 ERA) starts for the Giants against Dillon Gee (6-7, 4.45) on Tuesday. Zito began warming up in the bullpen in the bottom of the 16th in case the Giants tied it.