Forward Napheesa Collier (26 points) and guard Kayla McBride (28 points) spearheaded the effort, becoming the first pair of Lynx teammates to score at least 25 points apiece in a playoff game; both marks were also career playoff highs.
The Sun cut the Lynx's 16-point lead to three late in the fourth, but Minnesota warded them off down the stretch to stave off elimination.
"We've been in adverse situations all season and we've responded," McBride said. "Today was just another example of that."
After an 0-6 start in their first season without a member of their 2010s dynasty, the Lynx have an opportunity to win their first playoff series since 2017, when they won their most recent WNBA title. The franchise hadn't won a game in a playoff series since then.
"From 0-6 to 40 minutes from the semifinals," head coach Cheryl Reeve said. "What more could you ask for?"
Game 3 will be played Wednesday in Minnesota -- the best-of-three first-round series is a 2-1 format with the higher seed hosting the first two games and the lower seed hosting Game 3 if necessary -- at a time to be announced.
After a lackluster performance in Game 1 (14 points), Collier rebounded with a 26-point, 13-rebound double-double despite dealing with back pain that caused her to temporarily leave for the locker room. She described the issue as a "pinched nerve or an SI joint, I think it was ... nothing that's going to hurt you long term, but it sucks in the moment."
"I want to leave it all out there on the court and do whatever I can for my team," said Collier, an MVP candidate in her first full season back after giving birth last spring. "It's such a special group of women that I'm playing with this year and so I want to give them my all and I expect the same from them and I think that's why we have such a great team."
Connecticut faces its second first-round elimination game in two seasons; last year in the first round, en route to the WNBA Finals, it beat Dallas on the road.
"I think the biggest thing for us is not letting what happened today [happen again]," said Connecticut's Alyssa Thomas, who finished with 26 points, 8 rebounds and 6 assists. "We didn't come out ready to play. We knew what was at stake. And now unfortunately we've got to go play on their home court, which is a tough thing to do."
Lynx players said before the game that if they could steal a game on the road Sunday, they'd have the advantage going into Game 3.
"If we play like this again, we're so hard to beat, and it's fun to play that way," Collier said. "So, I can't wait for the next game."