The Packers had forged a 20-20 tie with 5:09 left in the fourth quarter when Mason Crosby capped an 11-play, 61-yard drive with a 24-yard boot.
But Colin Kaepernick shook off 4-degree temperatures and responded by guiding the 49ers 65 yards on 14 plays, including an 11-yard run on third-and-8, to set up Dawson's game-winning kick as time expired.
Kaepernick completed 16-of-30 passes for 227 yards with a touchdown and an interception, and ran for 98 more on seven carries as his 49ers eliminated Green Bay from the playoffs for a second straight year.
They will play at Carolina next week in the NFC divisional round.
Aaron Rodgers was limited to 177 passing yards with a score on 17-of-26 efficiency for the NFC North champion Packers.