Rissmiller, who scored only seven goals in the regular season, beat Tomas Vokoun with a shot from the edge of the left circle after taking a crisp pass from Patrick Marleau.
San Jose squandered a 4-2 lead after two periods, an advantage built with three goals in the middle frame after the Sharks lost top goal scorer Jonathan Cheechoo to an injured right knee.
But the Predators, who stood atop the NHL until March 29 before slipping to the West's No. 4 seed, scored twice in the final 6:55 of regulation to force the first overtime in the franchise's short post-season history.
Alexander Radulov scored his second goal with 7:05 left, and J.P. Dumont scored his second of the game with 50.4 seconds remaining in regulation, tipping a slap shot from Shea Weber past Evgeni Nabokov's glove.
Weber had two assists for a team that went 4-15-1 in the regular season when trailing after two.
Craig Rivet scored on a 5-on-3 power play and added an assist. Joe Thornton, the NHL's second-leading scorer with 114 points, and Marleau each had two assists for San Jose. Matthew Carle, Mike Grier and Milan Michalek scored a goal each.
San Jose has won five straight playoff games against Nashville after taking the final four in the first round last year.
The Sharks were outshot 20-16 through two periods and 43-38 overall, but they won without Cheechoo. The forward didn't return after Scott Hartnell crashed into him with a knee-on-knee collision nine minutes into the second period that bent Cheechoo's knee awkwardly. His status for the rest of the series was uncertain.
Radulov got Nashville back into the game almost by himself. The rookie in his first playoff game skated up the right boards, fought off Kyle McLaren for the puck, then scored over Nabokov's right leg at 13:05 of the third.
Nashville had a power play with less than four minutes left in regulation but couldn't score. With Vokoun pulled for an extra skater, the Predators kept the puck in their offensive end.
Martin Erat, who missed the final 11 games of the regular season, passed the puck out to Weber at the blue line. He one-timed the puck toward the net, and Dumont deflected it past Nabokov to create the loudest roar of the night in the sold-out arena.
With fans snapping plastic clappers handed out during intermission and into overtime, the Predators took six of the first eight shots and had several chances to end the game.
But Nabokov stopped each one, including a breakaway shot by Jordin Tootoo that he squeezed between his legs before falling onto his back.
Vokoun had his share of spectacular saves, including stopping Marleau on a breakaway 6:25 into the second overtime by stretching out to his left to block a backhander at the post.
But Vokoun could not stop Rissmiller when Marleau fed the puck to him for a shot that beat the goalie to the wide, right side of the net. Rissmiller celebrated by skating over and jumping up against the glass.
Notes: Thornton had nine points in 11 playoff games last year. ... Nashville came in having taken the ninth-fewest penalties in the NHL in the regular season and was the league's third-best penalty killers. San Jose finished 1-of-7 with the man advantage. ... San Jose scored nine of 17 goals with the man advantage when these teams met in the 2006 playoffs, including three 5-on-3s.