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It was an error-strewn start to the match from both sides, with Matt Hasselbeck, in for the injured Andrew Luck at quarterback for the Colts, throwing an interception with his very first pass attempt of the game.
The Falcons were unable to take advantage as Matt Bryant hit the upright with his 46-yard field goal, but they were gifted another chance to get on the board moments later when Frank Gore's fumble was recovered by Paul Worrilow - the same player who had intercepted Hasselbeck early in the contest.
Once again, though, the visitors came away unscathed as Matt Ryan's pass intended for Roddy White in the endzone was tipped into the arms of Dwight Lowery, who made the interception against his former club.
Full-back Patrick DiMarco would prove to be the unlikely hero in the first half for Atlanta, though, and he caught the first of two touchdowns right at the end of the opening quarter when Ryan found him wide open in the endzone.
Having not scored a touchdown all season before today, DiMarco doubled his tally for the match in the second quarter when he collected a short pass from Ryan at the front of the endzone for a one-yard score.
The Colts halved the deficit with a little under two minutes remaining in the second quarter, with Hasselbeck's short pass finding Ahmad Bradshaw, who was able to stroll into the endzone from seven yards.
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However, Atlanta, who have now lost four of their last five games, restored their 14-point advantage after the interval when Leonard Hankerson reached over having been picked out on the right by Ryan.
The Indianapolis comeback began at the end of the third quarter when Hasselbeck and Bradshaw combined for the latter's second score of the evening, although they missed a chance to draw level when the stand-in quarterback threw an interception moments after a fumble gifted them possession on the Atlanta 21.
Chuck Pagano's side would benefit from an interception shortly afterwards, however, as D'Qwell Jackson picked off Ryan's pass from the one-yard line before returning it the short distance for a levelling score.
Adam Vinatieri nudged the AFC South leaders ahead for the first time in the final minute of the match with a 43-yard field goal, and the Falcons did not have enough time to respond as the Colts held out for victory.
The hosts' third defeat on the bounce leaves them at 6-4 for the season, while the Colts improve to .500 with a 5-5 record.