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A closely-fought opening two quarters saw the 49ers take a narrow 6-3 lead at half-time.
Gore ran in a one-yard touchdown to give the home side the advantage early in the third quarter.
Dan Carpenter kicked a field goal for the Dolphins, before each side added another touchdown to leave Miami needing a touchdown to tie the game.
However, a 50-yard run from the San Francisco quarterback saw Kaepernick guide the 49ers to the third win in four games with him as starter.
Gore's performance saw the running back record 1000 rushing yards in a season for the sixth time in his career.
The 27-13 win keeps the 49ers top of the NFC West.