Alexei Ponikarovsky, Bryan Little, Dustin Byfuglien and Alexander Burmistrov also scored for Winnipeg (3-1-1), which was down 4-2 midway through the third period. The Jets have now won three games in a row.
Byfuglien's goal on the power play at 12:59 of the third period marked the first time this season the Islanders had let in a power-play goal in 17 short-handed situations.
Burmistrov's goal 65 seconds later tied the game 4-4 in front of a sellout of 15,004 at MTS Centre.
Jets backup goalie Al Montoya got his first start of the season against his former Islanders teammates and stopped 21 shots. He signed as a free agent with Winnipeg last July.
He played two seasons with the Islanders, but got into only 31 games last season because of a concussion suffered after a December hit by now-teammate Kane.
Nabokov faced 35 shots.
A turnover by the Jets led to New York's first goal, a wrist shot Ullstrom put past Montoya at 2:34 of the opening period. Ullstrom is on a four-game points streak with two goals and a pair of assists.
Winnipeg tied it up eight minutes later when Kyle Wellwood sent a pass across the front of the net to Ponikarovsky for his first goal with the Jets.
Winnipeg outshot the Islanders 12-6 in the first.
Little gave the Jets a 2-1 lead at 5:13 of the second period when he took a pass in front of the net from Nik Antropov and shot the puck under the stick of a stretched-out Nabokov.
Tavares went on a breakaway 22 seconds into the third period and tallied his first goal of the season with a high shot past Montoya to make it 2-2.
Tavares got his assist when his long shot was redirected by Moulson for a 3-2 lead at 5:09. Streit made it 4-2 with a power-play marker at 9:53.
Byfuglien's hard, low shot past a screened Nabokov got the Jets ignited at 12:59 and Burmistrov tied it at 14:04.
Winnipeg was 1 for 4 on the power play through regulation and the Islanders were 1 for 2.