Frans Nielsen struck a third-period hat-trick to deny the Stars, in an undulating contest at New York's Coliseum.
The Islanders led after the first period, with Anders Lee and Nikolay Kulemin goals coming either side of Vernon Fiddler's strike for Dallas.
Brock Nelson put the hosts 3-1 up early in the second stanza, but Dallas turned the game on its head in a 10-minute period - as goals to Tyler Seguin, Jason Spezza and Antoine Roussel gave them the lead.
Lubomir Visnovsky squared the match for the Islanders with their fourth just 30 seconds after Dallas had first taken the lead.
The third period was goalless for nine minutes before Nielsen broke the tie, assisted by Kulemin and Mikhail Grabovski.
His second - and Grabovski's second assist - put the Islanders two goals up inside the final five minutes, only for Dallas' Shawn Horcoff to score moments later and make it a one-goal game.
However, Nielsen's empty-netter as time expired sealed a clutch treble and a thrilling win for the hosts.
The Minnesota Wild also helped themselves to seven goals, as they chalked up consecutive wins with a 7-2 result over the Tampa Bay Lightning.
Despite only out-shooting the Lightning 22-19, the Wild were buoyed to victory by a double to Jason Zucker plus goals to Marco Scandella, Mikko Koivu, Erik Haula, Zach Parise and Jared Spurgeon.
The Buffalo Sabres ended a four-game losing streak, and extended the San Jose Sharks' run to four losses, with a 2-1 victory.
Cody Hodgson and Nicolas Deslauriers struck quick-fire second-period goals, which was enough to send the Sharks to another defeat and help Buffalo win for the second time this season.
The Chicago Blackhawks fell to successive losses, trumped by the St Louis Blues 3-2.
Max Pacioretty had one goal and an assist to help the Montreal Canadiens to their fourth straight win, a 3-1 result over the New York Rangers.
The Nashville Predators were felled in regulation for the first time this season, the Pittsburgh Penguins shutting them out 3-0.
Philadelphia Flyers goaltender Ray Emery made 35 saves to help his side to a 4-2 win over the Detroit Red Wings - who had 20 more shots than the eventual victors.
Oliver Ekman-Larsson struck an overtime goal for the Arizona Coyotes in their 2-1 win over the Florida Panthers, while Jaromir Jagr was the match-winner for the New Jersey Devils in their 3-2 OT triumph against the Ottawa Senators.
The Boston Bruins were denied a shutout in the final six minutes, but cruised to a 4-1 win over the Toronto Maple Leafs, while Joel Ward struck twice in the Washington Capitals' 3-1 victory over the Calgary Flames.