The 20-year-old Svechnikov beat Henrik Lundqvist for the game's first goal less than 5 minutes into the game. He followed with a power-play one-timer early in the second off a feed from Sami Vatanen, then buried a third at 14:02 of the third off a feed from Sebastian Aho — who assisted on all three of Svechnikov's goals.
"I think he'll get a few more (hat tricks) before his time's over in his career," Carolina coach Rod Brind'Amour said. "He's a gamer, too. this guy is one of those players in this break that worked on this game, he wanted to get better. I think you're seeing that."
The 6-foot-2 Svechnikov, the No. 2 pick in the 2018 draft, had an assist in Saturday's Game 1 victory then followed by delivering a game-high six hits in this one to bring a physical edge.
"I just love that style, just hit hard," Svechnikov said. "I think I'm playing a hard game. That's I think why I'm scoring goals."
Jordan Martinook also scored for the Hurricanes, following Svechnikov's second goal 71 seconds later for a 3-1 lead in a sequence that put Carolina ahead for good.
Petr Mrazek added 23 saves for the Hurricanes, including big glove stops on Brett Howden and Mika Zibanejad in the first period. The one on Zibanejad, a 41-goal scorer in the regular season, came with the Rangers having a chance to take a 2-1 lead late in the period.
The Rangers had swept the regular-season series, and the Hurricanes rarely led in any of those four meetings. But the Hurricanes have yet to trail in two games in Toronto and now have a chance to sweep the best-of-5 series.
Teams that have taken a 2-0 lead in a best-of-5 are 55-1 in those series, though the NHL hasn't had one since 1986.
"We've got to reset our minds and get focused on winning one hockey game," Rangers defenseman Marc Staal said. "That's all you can do at this point."
Artemi Panarin scored for the Rangers, beating Mrazek in the closing seconds of a 5-on-3 power play midway through the first. Lundqvist finished with 30 saves, but lost for the second time in the postseason against the Hurricanes after going 3-0 against them in the regular season.
Rookie goaltender Igor Shesterkin (10-2) didn't play for the Rangers. Quinn had been uncertain Sunday on Shesterkin's status after he was unavailable for Game 1. ... Carolina defenseman Dougie Hamilton didn't play Monday after missing Game 1 and multiple recent practices.