The visitors were trailing the Colts until the final few minutes when Jalen Hurts rushed for a seven-yard touchdown which was then converted by Jake Elliott to end the game with Philadelphia up 17-16.
The Eagles' quest for a perfect season ended with a shock 32-21 loss to the Washington Commanders last week, with victory in Indianapolis moving the side to 9-1 for the season so far.
The Colts failed to get much payoff from their dominance, with the home team relying on field goals for points aside from a touchdown by Pro Bowl running back Jonathan Taylor.
In other games, quarterback Kirk Cousins was sacked a career-high seven times as his Minnesota Vikings were hammered 40-3 by the Dallas Cowboys and the Atlanta Falcons kept their noses in front of the Chicago Bears 27-24.
In Massachusetts, the New England Patriots' Marcus Jones scored via an 84-yard punt return with just five seconds left for his side to beat AFC East rivals the New York Jets 10-3.
Down south, the Houston Texans lost at home to the Commanders 23-10 and the New Orleans Saints showed up reigning Super Bowl champions the Los Angeles Rams 27-20.
The Buffalo Bills beat the Cleveland Browns 31-23, the Baltimore Ravens went 13-3 over the Carolina Panthers 13-3 and the Detroit Lions defeated the New York Giants 31-18.
The Las Vegas Raiders were 22-16 overtime victors against the Denver Broncos and Joe Burrow tossed four touchdown passes for the Cincinnati Bengals to finish 37-30 over the Pittsburgh Steelers.
In the night's final game, the Los Angeles Chargers almost secured their own late win before Patrick Mahomes threw a 17-yarder to tight end Travis Kelce which put the visitors up 30-27 as the clock wound down.