Darren Sproles had seven receptions for 152 yards to go along with a touchdown run as the Eagles (2-0) won their first game on the road for the sixth consecutive season.
Indianapolis (0-2) has lost consecutive regular-season games for the first time since Andrew Luck and coach Chuck Pagano arrived in 2012.
The Colts led most of the night, including 27-20 late in the game. But Luck's late interception led to Foles' tying 6-yard TD pass to Jeremy Maclin, and after another three-and-out by the Colts, Foles drove the Eagles into position for the winning field goal.
"We felt we had some plays (in the first half), we just couldn't make them," Eagles coach Chip Kelly said. "But we're going to play 60 minutes and we played 60 minutes."
The Eagles' potent offence started slow, managing just two field goals in the first half and failing to score a touchdown before LeSean McCoy's 1-yard run with 4:32 left in the third quarter. They finished strong, tying it at 20 when Sproles managed to stay on his feet for a 19-yard run late in the third and again at 27 on a 6-yard TD pass to Jeremy Maclin.
Luck drove the Colts right back down the field and hooked up with Ahmad Bradshaw for a tiebreaking score, but Indy couldn't close it out and the Eagles took advantage.
Foles was 21 of 37 for 331 yards with the one score. LeSean McCoy ran 20 times for 79 yards and a touchdown.
Indy looked as if it would give Luck a respite from his late-game heroics. The Colts led most of the night, thanks to an unusually strong ground game.
But it just wasn't enough.
Luck tried to seal the game with a short pass to T.Y. Hilton with 5:08 to go and the Colts driving. But Malcolm Jenkins caught the ball on the run while it looked as if Hilton was grabbed on the play. There was no call.
Two plays later, McCoy was stopped for a loss but LaRon Landry was called for a horse-collar tackle. Sproles then took a swing pass for a 51-yard gain to the 6, and Foles found Maclin for the tying score.
The Eagles defence held again and Parkey finally won it.
"They made the plays they had to make. We just didn't play a 60-minute ballgame," Pagano said. "Turnovers killed us. Big runs, big plays in the pass game, killed us."
Indy controlled most of the first three quarters.
Luck wiped out a 3-0 deficit with a 1-yard TD pass to Bradshaw with 5:18 left in the first quarter. And after Parkey missed a 38-yard field goal, Luck set up Adam Vinatieri for a 46-yarder to make it 10-3 early in the second quarter.
After Parkey made a 23-yard field goal, Luck passed to Jack Doyle for a 1-yard TD with 1:13 left in the first half to make it 17-6.
When it looked as if the Colts might pull away following Vinatieri's 27-yard field goal midway through the third quarter, the Eagles rallied with the TDs from McCoy and Sproles.
Luck broke the tie with the TD pass to Bradshaw, and a late interception gave the Eagles one more chance to tie it.
"I just think we did a better job in the second half," Kelly said.
NOTES: Eagles CB Nolan Carroll II left the game with an injured elbow but returned in the third quarter after X-rays were negative. ... Philadelphia linebacker Mychal Kendricks left in the third quarter with a calf injury. ... Indy was hosting its first Monday night game since 2010.
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