After Franson bested Devan Dubnyk in tiebreaker's third round, Predators goalie Pekka Rinne made the advantage hold up by stopping Taylor Hall for Nashville's 10th win in 12 games against the Oilers.
Rinne turned aside 22-of-24 shots, improving to 9-1-0 in his last 10 games.
Alexander Sulzer scored his first career goal and assisted on Joel Ward's power play tally for Nashville.
Hall scored with less than 4 minutes remaining, forcing an overtime, and Andrew Cogliano also found the net for the Oilers, who fell to 2-12-2 in their last 16 games.
Dubnyk made 26 saves for Edmonton.