The 49ers were trailing 17-16 with 3 1/2 minutes left in the fourth quarter when Gore took Colin Kaepernick handoff, started left, cut back through the middle and ran to the Seattle 18-yard line before being forced out of bounds.
He made four more carries and Kaepernick ran for a first down before Dawson made good on the decisive field goal with 26 seconds left.
Eric Wright intercepted the Seahawks' Russell Wilson on the next possession to seal a third straight win for the 49ers (9-4), who pulled within two games of playoff-bound Seattle (11-2) in the NFC West.
Gore finished with 110 rushing yards on 17 carries. Kaepernick compiled 175 aerial yards with a touchdown and an interception.
Wilson completed 15-of-25 passes for 199 yards and a touchdown as the Seahawks had their seven-game winning streak snapped.