With scores level and time expiring, Cody Parkey slotted a 36-yard match-winning field goal - twice for good measure - at Lucas Oil Stadium in Indiana.
Parkey's first attempt, which split the uprights, was ruled out due to a time-out call, before he held his nerve to kick his third field goal of the contest from four attempts.
Eagles quarterback Nick Foles commandeered the comeback, completing 21 of 37 pass attempts for 331 yards and one touchdown pass.
Colts playmaker Andrew Luck gained just 172 yards from 20 completions, but did produce three TD passes.
Each side's number one rusher finished with near identical numbers, LeSean McCoy (Philadelphia) and Trent Richardson (Indianapolis) both gaining 79 yards.
McCoy had 20 rush attempts and also helped himself to one TD, while Richardson made 21 runs - his longest being 15 yards.
The Eagles have won their opening two games of the season, while the Colts dropped to a 0-2 record.
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