Zegras scored on a breakaway to cut it to 3-2 at 7:24 of the third, and Kulikov tucked in a wraparound to tie it 3-3 at 13:01.
Frank Vatrano then appeared to give Anaheim a 4-3 lead with 4:54 remaining, but Maple Leafs coach Sheldon Keefe challenged the play for goalie interference, and the call was reversed after a video review.
"We needed that, to come back on a good team like that and show resiliency," Gibson said. "Fighting back through the goalie interference and overturning that goal, finding a way to get two points in overtime, speaks a lot."
Toronto was playing its third road game in four days after losing to 4-3 in overtime to the San Jose Sharks on Thursday and 4-2 to the Los Angeles Kings on Saturday.
"We've got to get home and get rested," Keefe said. "Going to get some healthy bodies in our lineup here soon, too, so those kinds of things will give us a boost."
Matthews gave the Maple Leafs a 1-0 lead with a power-play goal at 4:23 of the first period. Nicholas Robertson brought the puck into the offensive zone and passed to William Nylander, who found Matthews skating through the middle for his third goal of the season.