Fantilli positioned himself at the bottom of the right circle and scored from a sharp angle with a one-timer off a pass from Charlie Coyle.
Columbus was on the power play after Matthew Knies drew a holding penalty against Shea Theodore just 15 seconds into the extra period.
Denton Mateychuk scored, and Cam Talbot made 26 saves, including 13 in the third period, for the Blue Jackets (4-5-0), who had lost two straight and four of five.
Tomas Hertl scored, and Carter Hart made 18 saves for the Golden Knights (8-4-1), who have lost three straight.
Mateychuk gave Columbus a 1-0 lead at 11:51 of the first period. He took a pass from Kent Johnson, who was in the left corner, and lifted a backhand over Hart while cutting toward the net.
Hertl tied it 1-1 at 17:13 of the first. He skated in and cleaned up a rebound in front after Talbot stopped a 2-on-1 chance from Braeden Bowman, who had cut back across the grain but had his backhand stopped by the goaltender's right pad.
