EJ Manuel put the Bills in front during the first quarter as he completed a yard run, leaving Dan Carpenter to kick for the extra points.
The Bills then doubled their lead minutes later, this time as Fred Jackson bustled through with a four-yard run to complete the touchdown and leave his side 14-0 up.
However, in an intriguing first quarter, the Falcons were able to earn themselves seven points as Steven Jackson completed a 27-yard run to bring the score to 14-7 going into the second quarter.
Carpenter added three more points after he scored a 29-yard field goal, bringing his side's tally to 17.
The Falcons pulled themselves back within touching distance just before the end of the second as Antone Smith completed a huge 38-yard run to bring the score to 17-13.
Smith's side did not stop there, however, as Bryant kicked a 49-yard field goal in the last minute to tie the score at 17-17 going into half time.
However, the Bills came right back at their opponents two minutes later with a touchdown of their own through Jackson.
Neither side could add to the score before the end of the third quarter, but the Bills retook the lead in the fourth when C.J. Spiller completed a 36-yard run.
However, with the match ticking towards its conclusion, Steven Jackson scored a touchdown to level the score and take it into overtime.
Bryant's 36-yard field goal then won it for the Falcons.