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That represents the fastest three goals scored by one team in a Stanley Cup Final game.
The Hurricanes forced overtime when forward Andrei Svechnikov scored on a six-on-four power play with 1:42 remaining in regulation.
They played without forward William Carrier, who suffered a second-period injury.
Vegas goaltender Carter Hart stopped 29 shots.
Defenseman Brayden McNabb returned to action wearing a protective facial cage after sustaining an injury in Game 2.
Two early second-period goals by Mark Stone and Jack Eichel were overturned following successful video challenges by Carolina.
The challenges were for an offside violation and goaltender interference, respectively.
"We just left our foot off the gas," said Shea Theodore.
"I think we have to be sharper in the third, but I liked the resiliency out of our group."
Theodore played on the top defensive pairing alongside McNabb. Team captain Mark Stone praised McNabb's quick return and work ethic.
"I wish I could say it's shocking, but it's not," said Mark Stone. "Just the way he comes to the rink every day, it's like nothing happened."
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The teams will have a two-day break before resuming the series for Game 4 on Tuesday night in Las Vegas.
