Vegas goaltender Carter Hart made 29 saves.
Two second-period goals by Mark Stone and Jack Eichel were overturned due to offside and goaltender interference, respectively.
This is the 10th time in NHL history that the first three games of the Stanley Cup Final were decided by one goal, last occurring in 2016 between Pittsburgh and San Jose.
Defenseman Brayden McNabb returned to the top pairing with Theodore, wearing a protective cage after a facial injury from a puck on Thursday.
"We just left our foot off the gas," Theodore said on the ABC broadcast.
"I think we have to be sharper in the third, but I liked the resiliency out of our group."
Captain Mark Stone praised McNabb: "I wish I could say it's shocking, but it's not.
Just the way he comes to the rink every day, it's like nothing happened. He's one of a kind."
Both teams have now blown two-goal leads in the series. Vegas rallied in Game 1, and Carolina responded with an overtime win in Game 2.
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The series resumes with Game 4 on Tuesday night in Las Vegas.
