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.