Dart, making his third NFL start after taking over from struggling veteran Russell Wilson, threw for one touchdown and ran for another while rookie running back Cam Skattebo rushed for 98 yards and three TDs as the Giants pulled away for just their second victory of the season.
The Eagles had been aiming to regroup after seeing their four-game winning streak to start the season end with a 27-17 loss to Denver on Sunday.
But the lack of offensive consistency exposed as they blew a fourth-quarter lead against the Broncos was in evidence again as they suffered back-to-back defeats for the first time since 2023.
After a back-and-forth first half, Philadelphia turned the ball over twice as they failed to score after the break.
New York's Cordale Flott intercepted a Jalen Hurts pass intended for Jahan Dotson and motored 68 yards to set up Skattebo's third touchdown, also thanks to a pass interference call in the endzone that set the Giants up at the one-yard line.
Hurts connected with running back A.J. Dillon with a pass for what would have been a first down, but Dillon fumbled and the Giants recovered midway through the fourth quarter.
Philadelphia's Saquon Barkley had run for 31 yards on his first two carries. However, that early Eagles' drive stalled and Philadelphia settled for a 42-yard field goal from Jake Elliott.
The Giants responded with a touchdown as Dart eluded a defender and sprinted 19 yards into the end zone to put New York up 7-3.
Dart hit Wan'Dale Robinson with a 35-yard touchdown pass that pushed the lead to 13-3 after Jude McAtamney's extra point attempt failed.
Philadelphia cut the deficit to three points with 1:49 left in the first quarter as Hurts led a 72-yard, seven-play drive capped by his three-yard touchdown toss to Dallas Goedert.
The Eagles then produced a 13-play, 74-yard go-ahead scoring drive, turning to their trademark "tush push" short yardage play four straight times from within the five-yard line.
Hurts piled over for a touchdown on the final attempt and Jake Elliott drilled the extra point to put Philadelphia up 17-13, but the Giants were up 20-17 at halftime thanks to Skattebo's four-yard touchdown run.
The Giants scored touchdowns on three of four possessions and gained 198 yards in the first half and Skattebo powered into the end zone early in the third quarter to extend the lead to 27-17, leaving the Eagles' hopes of a comeback completely lost.