The German giants, among the favourites to win the competition this season, were made to wait 28 minutes for their opener but soon found their stride.
Competition debutants Rostov, who comfortably beat Ajax to make it to this stage, restricted their opponents well in the opening quarter of the match as the chances were kept to a minimum.
Thomas Muller picked his spot when found by a teammate inside the box to mark his birthday with a well-taken goal, putting three-time Champions League-winning manager Carlo Ancelotti on his way to a triumphant start.
The third arrived via Kimmich 53 minutes in, with the youngster continuing his impressive form in front of goal by tucking home from a few yards out at the end of a flowing Bayern move.
Kimmich was at it again soon after, this time glancing home a header when getting forward, and there was almost a fifth of the night when Arturo Vidal saw his attempt flicked onto the crossbar.
Ancelotti's men, who have started their domestic campaign in perfect style with two wins from two, had further chances to extend their lead before Juan Bernat guided the ball home to round off the rout in added time.