Atlanta's Bobby Holik scored with 1:22 left in regulation to force the overtime.
Mike Comrie's second goal of the game gave New York a 4-3 lead early in the third period. Vasicek and Andy Hilbert also scored for the Islanders.
Eric Perrin scored on a penalty shot, and Ken Klee and Alexei Zhitnik also scored for the Thrashers.
The Islanders were dominant early, taking a 2-0 lead midway through the first period while holding a 9-1 advantage in shots on goal.
Thrashers goalie Kari Lehtonen partially blocked Comrie's first goal 9:15 into the game, but Lehtonen couldn't maintain control of the puck. An officials' review showed the puck fell back to the ice and crossed the line under Lehtonen's right knee.
Vasicek then scored 11:26 into the opening period. Lehtonen was caught leaning to his right, leaving half the net exposed as Vasicek crossed in front of the net for the shot.
The Islanders pushed the lead to 3-0 only 1:02 into the second period when Bryan Berard set up Hilbert's goal.
Atlanta fans began booing, but the Thrashers recovered with three straight goals.
The momentum turned on Atlanta's first penalty shot of the season. Perrin's scoring opportunity was spoiled when he was sent sprawling as he was pulled from behind by Berard, who was called for holding. Perrin was awarded the penalty shot, and he beat Rick DiPietro 3:53 into the period with a sweeping move that had DiPietro leaning the wrong way.
Klee, set up by newly acquired forward Colby Armstrong, scored Atlanta's second goal at the 13:46 mark, and Zhitnik's third goal of the season tied the game 3-all with 5:00 left in the second.
The Islanders regained the lead 3:17 into the third. Berard earned his second assist of the game on a pass to Comrie, who shot past Lehtonen into the right side of the net.
