Fantilli tied it 5-5 with Elvis Merzlikins pulled for the extra skater after his wrist shot went in off Red Wings defenseman Albert Johansson.
Kirill Marchenko and Kent Johnson scored in the shootout, and Merzlikins stopped both shooters he faced.
Marchenko and Johnson each had a goal, and Zach Werenski and Sean Monahan each had three assists for the Blue Jackets (13-9-5), who won their second straight after losing four in a row. Merzlikins made 28 saves.
Patrick Kane and Lucas Raymond each had a goal and an assist, and Moritz Seider and Andrew Copp each had two assists for the Red Wings (14-11-3), who have lost five of six. Cam Talbot made 25 saves.
Ivan Provorov gave Columbus a 1-0 lead at 17:55 of the first period before Dylan Larkin tied it 1-1 with a power-play goal at 1:09 of the second period. He redirected a Seider shot from between the circles.
Marchenko scored 12 seconds into a power play to put the Blue Jackets back ahead 2-1 at 5:18 on a shot from the left dot. It was his first game after missing the past four with a lower-body injury.
Johnson extended it to 3-1 at 7:11 on a rebound of a Cole Sillinger shot.
The Red Wings tied it 3-3 on power-play goals by Raymond at 9:08 and former Blue Jackets forward James van Riemsdyk at 10:20 during a double-minor for high-sticking to Dmitri Voronkov.
Fantilli’s power-play goal gave Columbus a 4-3 lead at 13:50, a one-timer under the bar from the right circle.
Kane tied it 4-4 at 9:35 of the third period with a wrist shot from the high slot.
Alex DeBrincat then gave Detroit a 5-4 lead at 11:38 with a shot straight-on from the high slot.