The Mets trailed 7-4 after six innings before Eric Young, Jr. stroked a two- run homer in the seventh off Alfredo Aceves to get the visitors within one.
Lucas Duda laced an RBI single off Matt Thornton (0-1) in the eighth to tie the game, and after Preston Claiborne replaced Thornton on the mound, Chris Young clubbed Claiborne's second pitch into the seats in left for a 9-7 lead.
After striking out Alfonso Soriano to end the seventh, Jenrry Mejia (4-0), who was sent from the starting rotation to the bullpen earlier Monday, worked around a pair of singles in the eighth to maintain the Mets' margin.
However, Farnsworth was able to secure his third save of the season by getting Brian McCann to ground into a game-ending 3-5-3 double play.
Curtis Granderson returned to Yankee Stadium for the first time since signing with the Mets in the offseason and slugged a two-run homer, while Eric Young, Jr. finished 3-for-5 with two runs scored in the Mets' fifth straight win over their cross-town rivals.
"I think tonight we just got some balls that we could drive," Mets manager Terry Collins said, referring to his team hitting four home runs. "(Yankee Stadium) is a great place for a right-handed hitter who can hit the ball to right-center field. This is a great ballpark for them."
Brett Gardner hit his second career grand slam in the setback, while Yangervis Solarte collected three hits, an RBI and two runs scored for the Yankees, who have lost three straight overall.
Jeter registered his first three-hit game since Sept. 30, 2012 in the loss.
"We would have liked to have won this game, especially having the lead late," Jeter said. "We just didn't score enough runs."
The Mets manufactured a run in the first on David Wright's RBI groundout that scored Eric Young, Jr. for a 1-0 lead, but Gardner's grand slam in the second gave the home team control.
The Yankees started the second with three consecutive singles, including Alfonso Soriano's 1,000th career American League hit, making him just the seventh player in MLB history to compile 1,000 hits in each league.
Mets starter Bartolo Colon briefly slowed the hit parade by retiring the next two batters, but Gardner jumped on a first-pitch fastball and scorched it into the seats in right for a 4-1 lead.
Travis d'Arnaud smacked a one-out solo shot in the fifth to trim the margin to two before Granderson followed Wright's leadoff single in the sixth with a drive over the wall in right-center field to tie the game.
The Yankees answered with three runs in the home half, as Soriano stroked a ground-rule double before scoring on Solarte's single to right.
Kelly Johnson then sliced a triple into the gap in left-center field to score Solarte, but the Yankees first baseman was caught in a rundown between third and home when Brian Roberts followed with a grounder to short.
After Gardner extended the inning with a single to right, Carlos Torres took over on the mound for the Mets. With Jeter at the dish, Gardner took off for second and d'Arnaud's throw sailed into center field, allowing Roberts to race home for a 7-4 lead.
Aceves replaced Yankees starter Hiroki Kuroda to start the seventh and walked d'Arnaud before Eric Young, Jr. smacked his first homer of the season two batters later to bring the Mets within 7-6.