The Giants took the opener 6-1 as Lincecum posted his first win of the season despite another uneven performance.
"It was a long day and a good day," Bumgarner said.
Slightly over 1,000 fans bundled up in the stands when the opener began on a windy, raw afternoon better suited for the Bay Area. The crowd later picked up, but Citi Field was nearly empty for the final out. A rainout Sunday forced the twinbill.
Sandoval hit a three-run homer in the first inning to back Bumgarner. The burly slugger has a 16-game hitting streak, tying the San Francisco record at the start of a season set by Mays in 1960, STATS LLC said.
"Willie Mays is one of the greatest," Sandoval said. "Everything's going well."
Hector Sanchez later hit his first big league homer, a two-run shot.
Bumgarner (3-1) allowed one run and three hits. He won his third straight start, a streak that began shortly before the 22-year-old lefty signed a $35.56 million, six-year contract.
"I kind of liked watching hitters and seeing what they did against Timmy," Bumgarner said.
New York loaded the bases with one out in the eighth against Javier Lopez. But reliever Clay Hensley retired pinch-hitter Jordany Valdespin on a popup -- on the first pitch he saw as a major leaguer -- and struck out pinch-hitter Ike Davis looking.
Frustrated, Davis flung his bat and later threw his helmet. He made a key error in the first game and went 0 for 4, striking out twice, grounding into a double play and bouncing out with the bases loaded.
San Francisco won for the eighth time in 11 games and swept a doubleheader from the Mets for the first time since 1997. Then again, teams don't play many doubleheaders anymore.
The Mets have lost five of six and dropped to .500 for the first time this year.
"They're a tough team to pitch against. They run all those switch-hitters at you. They just outhit us and outpitched us tonight. It was pretty simple," Mets manager Terry Collins said.
Dillon Gee (1-2) gave up seven runs and a career-high 12 hits in 6 2/3 innings.
Both games were similar: Sandoval put San Francisco ahead for good in the first inning, the Giants supported their starter with two home runs, and they got plenty of pitching.
Battered in his first three starts, Lincecum (1-2) allowed one run in five innings and struck out eight, often throwing his fastball past the Mets. The two-time Cy Young Award winner also walked five and gave up four hits in lowering his ERA from 10.54 to 8.20.
Lincecum admits he'd been overthinking his recent outings.
"Your head's spinning and you're going to 50 different answers trying to get one," he said.
"We get caught up in negatives so easily," he said, "and forget about the positives."
Lincecum got help on his 108th and final pitch. With the bases loaded, one out and the Giants ahead by five runs, Davis hit a grounder up the middle. Second baseman Emmanuel Burris stopped the ball and made a backhanded flip with his glove to shortstop Brandon Crawford, who barehanded the toss and turned a terrific double play.
Miguel Batista (0-1) lasted 3 2/3 innings in a spot start. The 41-year-old righty began the game by pitching out of the stretch -- fittingly, the Giants had runners on base against him all day.
Sandoval hit an RBI single in the first, but was thrown out trying to stretch it into a double. Posey followed with his home run.
Schierholtz hit a three-run homer in the third, set up by Davis' error at first base with two outs and none on.
Mets LF Jason Bay bruised his ribs trying to catch Gregor Blanco's two-run double in the second game. The ball bounced out of Bay's glove and he landed face-first on the grass near the warning track. X-rays were negative and Bay will be re-examined Tuesday. ... Marlins SS Jose Reyes makes his return to Citi Field on Tuesday night when Miami visits the Mets. "He was a dynamic, exciting player for us. I think he deserves a warm reception from our fans," Mets GM Sandy Alderson said. "I hope he gets a good reception." ... Johan Santana (0-2, 3.97 ERA), pushed back a day by the rainout, starts for the Mets on Tuesday against Marlins ace Josh Johnson (0-2, 5.94). ... Matt Cain (1-0, 1.88) starts for the Giants on Tuesday night at Cincinnati. ... Giants 2B Freddy Sanchez (dislocated right shoulder) begins a rehab assignment Monday night for Class-A San Jose. Giants manager Bruce Bochy said the team had mapped out an 18-day rehab for Sanchez. ... Giants 1B Aubrey Huff did not play in either game. He's in a slump, and also had a key defensive lapse Saturday while playing 2B for the first time in his career.