Rackers had also sent the game into overtime with a successful kick less than 6 minutes beforehand.
Kurt Warner completed 36 out of 53 passes for 361 yards and three touchdowns for Arizona (7-8), which broke a two-game losing streak.
Anquan Boldin also caught 13 passes for 162 yards and a pair of touchdowns in the victory.
Chris Redman was 28-for-42 passing with 315 yards, two touchdowns and an interception for the Falcons (3-12), who lost their sixth straight game.
Roddy White caught 12 times for 141 yards and Laurent Robinson caught seven passes for 114 yards.
Morten Andersen was also 2-for-2 on field goals in the loss.