Higgins did some quick stick-handling on his way toward Jets goalie Ondrej Pavelec and flipped the puck by him after the netminder went down.
Vancouver (30-28-10) is now 2-4-1 in their last seven games.
The game was a big improvement for the Canucks, who collapsed on Monday and gave up seven goals in the third period in a 7-4 loss to the New York Islanders.
The Jets (30-28-9) are 0-2-3 in their last five.
Alexandre Burrows, with the first and second goals of his injury-marred season, tied the game 2-2 at 11:17 of the third period.
Jets defenceman Zach Bogosian was trying to get the puck out from behind the net, but the puck flipped up high to Burrows at the front side and he batted it in past Pavelec.
It was only Burrows's 36th game of the season.
Andrew Ladd scored his 18th goal of the season and Michael Frolik added his 13th that put the Jets ahead 2-1 early in the third period.
Pavelec stopped 30 of the 32 shots he faced in regulation and overtime, while Eddie Lack made 31 saves for Vancouver.
The Canucks now have 70 points and Winnipeg 69 as they battle for a playoff spot in the Western Conference. Dallas holds the last wild-card berth with 74 points. Phoenix has 73. The Stars and Coyotes were idle Wednesday night.
Vancouver outshot Winnipeg 14-9 in the scoreless first period, with the Canucks getting three power-play opportunities and the Jets two.
Winnipeg scored the game's first goal on a nice passing play by Bryan Little.
With Frolik skating toward the net and Ladd behind him, Little sent his pass across to Ladd and he fired a high shot past Lack at 3:48 of the second period.
Jets winger Evander Kane got a penalty shot midway through the period after Christopher Tanev grabbed him on the way to the net, but his high shot over Lack's glove missed the net.
Burrows scored after teammate Jordan Schroeder won the faceoff and Burrows caught Pavelec out of position with 4:22 left in the period.
Canucks centre Ryan Kesler was in a knee-to-knee collision with Jim Slater and didn't return for the third period.
Frolik scored his 13th of the season as he raced to the right side of the net and took a pass from Olli Jokinen to make it 2-1 at 2:18 of the third.
Burrows knotted the score 2-2 at 11:17 of the third, a period in which the Jets outshot the Canucks 16-5.
Vancouver was 0 for 5 on the power play and the Jets were 0 for 4.
Notes: Henrik Sedin was marking a major milestone, playing in his 1,000th NHL game, all with Vancouver. He was drafted third overall by the Canucks in the 1999 NHL Entry Draft.