The No. 1 overall pick in this year's draft also scored on a 5-yard run and scrambled up the middle of the field 20 yards on third-and-19 to prolong the third game-winning drive of his young career.
Lavonte David sealed the fourth win in six games for the Bucs (6-6), intercepting Matt Ryan's pass intended for Julio Jones on Atlanta's final possession. The Falcons (6-6) have lost five straight, six of seven overall, after beginning the season 5-0.
Winston completed 18 of 27 passes for 227 yards, one touchdown and one interception.
But it was the rookie's timely run and ability to gain another 10 yards after appearing to be stopped at the Atlanta 33 that typified what the 2013 Heisman Trophy winner has meant to Tampa Bay, which has overcome a slow start to contend for a playoff berth.