Playing the first game of a two-stadium split doubleheader and the final Subway Series game at the current Bronx stadium, Delgado gave the Mets a healthy cushion when he belted his 12th grand slam off Ross Ohlendorf in the sixth inning.
Delgado also had a two-run double in the fifth inning that put the Mets up by two runs, and then completed his barrage with a three-run shot in the eighth off LaTroy Hawkins.
"Every time he steps up to the plate, you know he can do damage," Yankees outfielder Johnny Damon said. "The players still see him as a guy who's scary, regardless of what kinds of numbers he puts up.
"He's in the heart of that lineup, and it makes a big difference, whether he's hitting .220 or he's the MVP-caliber [player] from a few years ago."
Returning home after spending most of their night watching torrential downpours and an impressive lightning show in Pittsburgh, the Yankees had trouble keeping up, even despite a staggered report time to help with the quick turnaround.
Rookie right-hander Dan Giese made his second Major League start and was roughed up for six runs on five hits through four-plus innings. Giese (1-3) exited after 95 pitches, leaving a bases-loaded, none-out jam for Edwar Ramirez.
"It's embarrassing," Giese said. "That's my fault, and I pretty much set the tone for the game. When I come out of the game with the bases loaded and nobody out, that's tough on any bullpen. Top to bottom, they're just a good hitting team. Those guys made names for themselves for a reason.
"If you throw the ball not on the corners, especially a guy like me, you're going to get hit."
Carlos Beltran -- who earlier belted a two-run homer, his 12th -- grounded into a run-scoring fielder's choice to tie the game at 4, and Delgado followed by rattling a two-run double into the right-field corner.
"Danny battled and they put some long counts on him, and he threw a lot of pitches in his four-plus innings," manager Joe Girardi said. "But you felt that if we held them down, we'd have a chance to come back. We weren't able to do that, and Delgado killed us today. It's frustrating."
David Wright added a run-scoring single, one of his four hits on the afternoon, facing Ohlendorf in the sixth to pad the advantage. Ohlendorf took the worst of the damage, charged with six runs, including two of the runs that scored on Delgado's second home run.
The reliever was optioned to Triple-A Scranton/Wilkes-Barre after the game to make room for left-hander Kei Igawa, who was to join the Yankees at Shea Stadium.
"I felt like I had a stretch early on where I had four or five good outings in a row, and I was hoping to get started on another one of those," Ohlendorf said. "Hopefully the next one will be better. I do feel like I'm throwing the ball better than I have been in some ways. I just need to throw my slider for strikes."
Girardi said he still believes in the Yankees' pitching staff.
"Obviously, today was a tough day," Girardi said. "Dan got some balls over the plate that they hit, and the walks hurt us bad today. Our bullpen was not able to shut them down today. Our bullpen has been pretty good, but today we were not able to shut them down."
The Yankees were swept in the three-game Yankee Stadium portion of the Subway Series.
"We definitely don't feel good about it," Damon said. "We've faced some pretty good pitchers."
Their Interleague slate completes with three games across the Triboro Bridge at Shea Stadium this weekend. That includes the second half of the doubleheader on Friday evening.
Sidney Ponson will make his 2008 Yankees debut against Pedro Martinez, having been added to the roster when the club placed outfielder Hideki Matsui on the 15-day disabled list between games.
In the afternoon game, Mike Pelfrey was not stellar, but he proved effective. The Mets right-hander completed five innings with four runs and eight hits charged to his tally, walking four and striking out five.
Pelfrey (5-6) worked out of serious trouble in the fourth inning, when the Yankees loaded the bases on a hit, a walk and a Fernando Tatis error when Pelfrey couldn't catch a throw to first base. Jorge Posada flied out to end the inning, stranding three aboard, and the Yankees also left two on in the fifth after a walk and error.
Jason Giambi had a run-scoring, broken-bat single to put the Yankees on the board in the first inning, and Alex Rodriguez, Robinson Cano and Wilson Betemit collected RBIs in a three-run third. Rodriguez hit his 15th home run, a long solo shot off Pedro Feliciano in the sixth, as part of a three-hit afternoon.
Derek Jeter's fourth-inning double extended his hitting streak to 14 games and was also the 400th two-base hit of his career, one night after a double was washed away by New York's rainout. Jeter has also hit safely in 18 straight games against the Mets, and has reached base in 40 consecutive Interleague games.
The game was the Mets' first victory in a two-stadium split doubleheader against the Yankees. The two teams played the first one on July 8, 2000, with the Yankees winning both games, 4-2 -- marked by Dwight Gooden's win in Game 1 and Roger Clemens drilling Mike Piazza in the helmet in Game 2. The second contests came on June 28, 2003, when the Yankees defeated the Mets by scores of 7-1 and 9-8.
Bryan Hoch is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.