Rinne -- a Vezina Trophy finalist for the NHL's best goaltender -- allowed two soft goals, the first when Jets defenseman Tyler Myers' shot went off Rinne's blade and in for a goal at 8:41 of the first period.
Jets center Paul Stastny made it 2-0, finding the back of the net at 10:47.
Predators coach Peter Laviolette pulled Rinne, putting backup Juuse Saros in the net. It was the third time in the series -- all in Nashville -- that Saros came on in relief for Rinne.
"The biggest moment of the season, it's a terrible feeling,'' Rinne said. "You let your teammates down, and that's what happened tonight. That's tough to swallow.''
Saros did not fare better than Rinne, giving up three goals on the night.
Laviolette called the goals given up by Rinne "fluky.''
"There's not a lot of things that you can do if a game's not going your way if you can't use your timeout,'' Laviolette said. "It was way too early to use a timeout at that point. Again, I just tried to send the game in a different direction, put a pause in the game and send it in a different direction.''
Rinne's 10:07 in the net was the least amount of time played by a starting goalie in a Game 7 in NHL postseason history, according to ESPN Stats & Information.
Stastny and Mark Scheifele each finished the game with two goals for the Jets. Scheifele's postseason tally now stands at 11, the most in the NHL. He had seven of those goals in Nashville.
Connor Hellebuyck -- who is also a Vezina finalist -- made 36 saves for Winnipeg. He said seeing Rinne get pulled was "heartbreaking."
Defenseman P.K. Subban scored the lone goal for the Predators on the board with a first-period power-play goal.
The Predators, winners of the Presidents' Trophy for best team in the regular season, lost to Pittsburgh in the Stanley Cup Final last June. Nashville is now the the ninth Presidents' Trophy winner in 10 years not to win it all.
The Jets will take on the Vegas Golden Knights in the Western Conference finals for a shot at the Stanley Cup, with Game 1 set for Saturday in Winnipeg.
The Associated Press contributed to this report.