Brock Nelson, who took Tavares' place on New York's first line, outworked three Toronto players in the corner and slid a pass toward the slot. James van Riemsdyk tried to clear it but deflected the puck right to Visnovsky, who snapped a shot past Jonathan Bernier for his third of the season.
It was the last in a series of sloppy plays by the Maple Leafs that included two shorthanded goals by New York's Michael Grabner during the same Toronto power play.
"We gave them three goals. You can't win in the NHL giving away three goals," Toronto coach Randy Carlyle said. "Gifts. Total gifts. We were fortunate to get a point."
The victory ended New York's six-game losing streak at the Coliseum and came in their first game without Tavares, who will miss the rest of the season after injuring his left knee while playing for Canada at the 2014 Sochi Olympics. At 9-14-8, the Islanders have the worst home record in the NHL.
With 54 points, the Islanders' playoff hopes are all but dead, but coach Jack Capuano said his team has no intention of rolling over.
"I still believe, and it's still one singular goal; one game at a time, one shift at a time," he said.
Rookie Anders Lee scored his second goal of the game with 2:40 left in regulation to force overtime and cap a wild third period that saw the teams combine for five goals in a span of 8:30.
The Maple Leafs trailed 2-1 after two periods but tied the game 8:50 into the third when defenseman Paul Ranger tapped the carom of van Riemsdyk's shot off the post past Nabokov for his third of the season. Dion Phaneuf put Toronto ahead at 11:26, blasting a one-timer from the right circle past Nabokov after a turnover by Islanders defenseman Travis Hamonic.
But New York tied it at 12:52 on the first goal by Lee, one of three players called up by the Islanders from their American Hockey League affiliate, the Bridgeport Sound Tigers, due to injuries. In addition to Tavares, the Islanders were missing second-line center Frans Nielsen, who's sidelined with a broken hand, and forward Matt Martin.
"They were a big factor in why we won the game tonight," coach Jack Capuano said of Lee, Ryan Strome and Mike Halmo. "Lee's as advertised, a big guy who goes to the net. … Halmo was great, a good energy guy for us who finished a lot of checks tonight and made some skill plays. Strome made a nice play on the power play too."
Joffrey Lupul put Toronto back in front 62 seconds later, but Lee tied it again when he deflected the puck past Bernier after Strome tried to bank the puck into the net off the goaltender.
"Getting around the paint and in the goalies eyes is something I've always tried to do," Lee said. "I've got the body to do it and that's my job out there."
Carlyle said his team didn't match the Islanders' compete level.
"Mentally, we looked like we were going to out-skill the club we were playing against," Carlyle said. "They tried to grind it with us, and we didn't play anywhere near the grind game that we're capable of. We tried to out-skill that club tonight.
"We had two one-goal leads in the third period. Those are the frustrating parts right now. "We'll just go back to work tomorrow."
Phil Kessel, who had the second assist on Ranger's goal, scored in the first period. Bernier made 30 saves for Toronto (32-22-7), which is 11-2-2 in its past 15 games and holds the first of the two wild-card playoff berths in the Eastern Conference. Nabokov stopped 18 shots for New York.
Kessel, the leading scorer for the United States in Sochi with five goals, gave Toronto the lead 6:53 into the game. He drifted into the slot, took a feed from Tyler Bozak and snapped a shot off the far post and into the net for his 32nd of the season. Kessel is second in the League in goals behind Alex Ovechkin of the Washington Capitals, who has 41.
Grabner, who scored five goals in four Olympic games playing for Austria, put the Islanders in front late in the period by scoring twice in 48 seconds during the same Toronto power play.
Grabner tied the game at 15:53, finishing off a pass from Casey Cizikas, who had picked the pocket of Toronto defenseman Jake Gardiner. He put New York ahead at 16:01 when Bernier fielded Andrew MacDonald's clearing pass but bounced his pass off the skate of defenseman Morgan Rielly; the puck came right to Grabner, who quickly put it into the unguarded net.
The two goals gave Grabner 11 for the season and matched his output in 30 previous games at Nassau Coliseum this season. The last Islander to score two shorthanded goals on the same power play was Zigmund Palffy, on April 17, 1999.
"We kept coming and obviously had a big shift to score that tying goal," Grabner said. "Everyone fed off each other tonight. We still have a lot of games left. Hopefully we can keep it up."
Neither team mustered a lot of offense during a scoreless second period. Nabokov made his best save midway through the period when he stopped van Riemsdyk's backhander on a 2-on-1 break. Bernier stopped Grabner's wide-open wrister a few minutes later.
Material from team media was used in this report.