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 a right knee injury.
But the Predators 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 slapshot 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 No. 2 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.