The win lifted the Predators past the Detroit Red Wings and into fourth place in the Western Conference. The loss officially eliminated Dallas from postseason contention.
Patric Hornqvist provided insurance by scoring with exactly two minutes remaining in the third period for the Predators, who hold a one-point lead over Central Division-rival Detroit with one game remaining. The Red Wings, who dropped a 2-1 home decision to New Jersey earlier on Thursday, will host fellow Original Six rival Chicago on Saturday while Nashville will visit Colorado.
The win also clinched a playoff spot for the Los Angeles Kings, who faced San Jose later Thursday.
After two scoreless periods, Bouillon finally opened the scoring 4:28 into the third. Nashville established pressure in the offensive zone before Bouillon's backhander beat Kari Lehtonen for his fourth goal - with all of them coming since March 12. The blue-liner is just one tally shy of his career-high.
Rinne, who missed Tuesday's game with a virus, made the modest offense stand up by stopping all 12 shots in the third period for his fifth shutout and 25th career.
Lehtonen, who finished with 29 saves, threw his stick into the corner in disgust after he yielded Hornqvist's goal.