The defending Super Bowl champions trailed 12-0 early and were down 19-14 in the fourth quarter before Carson Wentz made key throws and undrafted rookie Josh Adams delivered big runs.
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Adams scored on a 1-yard touchdown run and ran up the middle for the 2-point conversion to put Philadelphia ahead 22-19.
After the Giants tied it on Aldrick Rosas' third field goal, a 29-yarder, the Eagles controlled the ball for 5:27.
Coach Doug Pederson went for a fourth-and-1 at the Giants 42 and Wentz completed a 12-yard pass to Nelson Agholor right before the two-minute warning. Adams ran three times and Elliott made the go-ahead kick.
The Giants started at their own 34 with 16 seconds left but couldn't do much.
The Eagles (5-6) stayed in the mix in a mediocre NFC East. The Cowboys and Redskins are tied for first place at 6-5. The Giants fell to 3-8.
In the first half, Eli Manning picked apart an injury-depleted Eagles defense that was missing its top four cornerbacks, a starting safety and starting linebacker. But Manning threw an interception to Malcolm Jenkins at the Eagles 2 before the second quarter ended.
Saquon Barkley had 101 yards rushing for New York, and Odell Beckham Jr. caught only five passes for 85 yards against backups.
The Eagles relied on their little-used rushing attack to take the lead in the fourth quarter. Wentz tossed a 23-yard screen pass to Corey Clement and the next six plays were runs, capped by Adams' score.
Adams had a career-best 84 yards on 22 carries.
After Barkley broke loose up the middle for a 51-yard TD to give the Giants a 19-3 lead, the Eagles finally answered.
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Wentz fired a 15-yard TD pass to Zach Ertz and flipped the ball to Clement on an option play for the 2-point conversion, cutting the deficit to 19-11 late in the second quarter.
Elliott's 28-yard field goal cut it to 19-14 in the third.
The Eagles stepped up on defense, forcing three straight punts in the second half to allow the offense to rally.
Coming off the most lopsided loss by a defending champion - 48-7 at New Orleans - the Eagles fell behind early again.
The Giants jumped ahead 6-0 on the opening drive. Manning flipped a 13-yard TD pass to Barkley, but the 2-point conversion failed after a penalty on the Eagles allowed the Giants to go for it from the 1.
The Eagles appeared to tie it on a 52-yard TD run by Adams but a holding penalty on center Jason Kelce negated the score.
PHILLY DOMINANCE
The Eagles are 17-4 against the Giants since 2008, including 9-1 in last 10.
NO-NAME SECONDARY
The Eagles used Cre'von LeBlanc, Chandon Sullivan, and De'Vante Bausby as their starting cornerbacks. Only LeBlanc had ever started a game in the NFL.
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SLOW STARTS
The Eagles were scoreless in the first quarter for the fifth straight game and ninth time this season. They only have 21 points in the opening quarter.
INJURIES
Giants: TE Evan Engram injured his hamstring in warmups and didn't play. WR Quadree Henderson left with a shoulder injury.
Eagles: CBs Jalen Mills, Sidney Jones and Avonte Maddox and LB Jordan Hicks didn't play. CB Ronald Darby and S Rodney McLeod already were lost for the season.
UP NEXT
Giants: Host the Chicago Bears (8-3) on Sunday.
Eagles: Host the Washington Redskins (6-5) on Dec. 3.
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