Rinne had 43 victories this season, a franchise record, and held the potent Detroit offense in check to begin the post-season battle.
Bourque put the Predators in front for good with 7:31 to play in the second period and gave Nashville a 3-1 cushion with 8:25 left in regulation.
Tomas Holstrom brought Detroit within one with 2:07 left, but Rinne was held on the rest of the way to improve his post-season record to 9-10.
Jimmy Howard stopped 23 shots for the Red Wings, who finished as the No. 5 seed in the Western Conference and would have had home-ice advantage against Nashville if they had defeated Chicago on the final day of the regular season.
Nashville will host the second game of the series on Friday.
Ray Liotta sues skin care company over use of likeness
Kim Kardashian, Kanye West reportedly set wedding date