Josh Beckett (2-1) helped himself with an RBI single in his second straight victory after 14 winless starts, but the fiery right-hander left frustrated after failing to get an out in the sixth inning.
"I was mad at myself," Beckett said. "Last thing you want to do is kill your bullpen the first game of a series."
"Tonight was great," Gonzalez said. "We took walks, putting pressure on and putting up runs. We tallied up a pretty easy win there at the end."
Dodgers third baseman Juan Uribe limped off with a right hamstring injury after his ninth-inning double. Puig, who had three hits and scored twice to extend his recent tear at the plate, delivered an RBI single during a three-run ninth.
Curtis Granderson (three hits) and Lucas Duda homered in the sixth for the Mets, who dropped to 5-13 in May with their fifth defeat in six games. Twice they loaded the bases with one out and came up empty in the opener of a nine-game homestand.
New York (20-24) fell a season-worst four games under .500 and lost for the 10th time in 11 meetings with the Dodgers dating to July 1, 2012.
The teams combined to use 12 pitchers in a yawner that took 4 hours, 8 minutes -- a minute short of tying the Mets record for a nine-inning game.
"It seemed like we were on the field the majority of the game and that's not good," New York third baseman David Wright said.
Puig drew a leadoff walk in the fifth and Gonzalez hit his 11th homer into the Dodgers' bullpen in right-center. The streaking slugger has eight hits in his past 12 at-bats, including a home run in Sunday's loss at Arizona.
"I'm just staying with it all season long. Not going to change a thing," Gonzalez said.
Matt Kemp doubled and Crawford's RBI single chased Montero, who arrived in the majors with a reputation for fine control. One of the team's top pitching prospects, the 23-year-old right-hander has walked six and served up three home runs in 10 1/3 big league innings.
"I'm going to find out what he's trying to do. If he's trying to pitch to the corners, in this league you've got to earn that, you've got to earn the corners," Mets manager Terry Collins said.
"Once the umpire knows you'll pound that strike zone, he'll give you some stuff. He fell behind a lot tonight. That's why he threw a lot of pitches. We've got to get him to realize he's good enough to get strike one and go from there."
Uribe fisted a run-scoring single over a drawn-in infield against Jeurys Familia to cap the outburst.
Puig, the co-NL player of the week, doubled off the top of the right-field wall in the sixth and scored on Hanley Ramirez's single to make it 6-1.
Earlier in the game, the exuberant right fielder appeared to lose track of the outs when he fired a throw to third base after catching a fly ball to end the third inning. That drew chuckles from Puig's teammates -- and a long stare from a perplexed Uribe.
As he waited to be removed by manager Don Mattingly, Beckett bounced the ball on the back of the mound and appeared to be venting to teammates about a pitch "right down the middle."
"He hit a wall, which is unusual for him," Mattingly said. "It happened quick."
Dodgers LHP Hyun-Jin Ryu (shoulder inflammation) is scheduled to come off the disabled list Wednesday night to make his first start since April 27. Ryu has not allowed a run in 26 innings on the road this season, going 3-0 in four starts. ... Beckett entered with a 1.67 ERA in his previous six outings. Pitching at Citi Field for the first time, he improved to 7-2 in 15 career starts against the Mets. It was the first time he faced them since 2009 with Boston. ... Mets CF Juan Lagares was home in the Dominican Republic because of a death in the family.