Robert Lewandowski's penalty gave the German champions the lead before Thomas Muller swept home on his 27th birthday.
Joshua Kimmich stabbed in and quickly added a header as Bayern boss Carlo Ancelotti became the first man to manage seven clubs in the competition.
Juan Bernat fired in from 10 yards for 5-0. Atletico Madrid won 1-0 at PSV in Group D's other game.
Kimmich, 21, has now scored his first goal for Germany, first Bayern Munich league goal and first Champions League goal in a 10-day period.
After beating the Russian side, Ancelotti, who has won the tournament three times as a manager, said: "Can we win the Champions League? For sure our team will be competitive this season and for the moment everything is good, but the first objective is to win the group, and then we will see the draw."
English referee Martin Atkinson ruled out "a good goal", according to PSV boss Phillip Cocu, after Luuk de Jong's fifth-minute effort was disallowed in Eindhoven.
Saul Niguez's scissor kick from 18 yards proved decisive for Atletico, who had goalkeeper Jan Oblak to thank for saving Andres Guardado's penalty on the stroke of half-time.
In Group B, Anderson Talisca produced a stunning 93rd-minute free-kick to earn Besiktas a 1-1 draw against his parent club Benfica.
"It's a draw that feels like a victory," said Besiktas midfielder Ricardo Quaresma.
Talisca's strike in Lisbon leaves both sides adrift of Napoli, who came from behind as summer signing Arkadiusz Milik struck twice in nine minutes to overhaul Dynamo Kiev's lead before half-time in Ukraine.
The Italian side netted with their only two shots on target against Kiev goalkeeper Oleksandr Shovkovski, who became the second-oldest player in Champions League history at 41 years and 255 days. Ex-Lazio player Marco Ballota remains the oldest at 43 years and 252 days.
Bulgarian side Ludogorets - in the Champions League group stage for the second time - faced 14 shots at Basel but left Switzerland with a 1-1 draw, matching the other result in the group between Paris St-Germain and Arsenal in Paris.
Barcelona produced the stand-out result of the night as Lionel Messi struck a hat-trick in a 7-0 win over Celtic, while the other fixture in Group C between Manchester City and Borussia Monchengladbach has been rescheduled for Wednesday due to adverse weather.
Match ends, FC Bayern München 5, FC Rostov 0.
Second Half ends, FC Bayern München 5, FC Rostov 0.
Attempt missed. David Alaba (FC Bayern München) left footed shot from outside the box is close, but misses to the right from a direct free kick.
Renato Sanches (FC Bayern München) wins a free kick in the attacking half.
Foul by Aleksandru Gatcan (FC Rostov).
Goal!  FC Bayern München 5, FC Rostov 0. Juan Bernat (FC Bayern München) left footed shot from the centre of the box to the centre of the goal. Assisted by Franck Ribéry.
Franck Ribéry (FC Bayern München) wins a free kick on the left wing.
Foul by Aleksandru Gatcan (FC Rostov).
Attempt blocked. Franck Ribéry (FC Bayern München) right footed shot from outside the box is blocked. Assisted by Robert Lewandowski.
Joshua Kimmich (FC Bayern München) wins a free kick in the defensive half.
Foul by Denis Terentjev (FC Rostov).
Corner,  FC Bayern München. Conceded by Vladimir Granat.
Attempt saved. Aleksandr Erokhin (FC Rostov) header from the centre of the box is saved in the top right corner. Assisted by Timofei Kalachev.
Foul by Arturo Vidal (FC Bayern München).
Moussa Doumbia (FC Rostov) wins a free kick in the attacking half.
Offside, FC Bayern München. Arturo Vidal tries a through ball, but Franck Ribéry is caught offside.
Attempt missed. Timofei Kalachev (FC Rostov) right footed shot from the right side of the box is too high. Assisted by Aleksandr Erokhin.
Corner,  FC Rostov. Conceded by Javi Martínez.
Javi Martínez (FC Bayern München) wins a free kick in the defensive half.
Foul by Moussa Doumbia (FC Rostov).
Attempt blocked. David Alaba (FC Bayern München) left footed shot from outside the box is blocked.
Franck Ribéry (FC Bayern München) wins a free kick in the attacking half.
Foul by Aleksandru Gatcan (FC Rostov).
David Alaba (FC Bayern München) wins a free kick on the left wing.
Foul by Aleksandr Erokhin (FC Rostov).
Foul by Joshua Kimmich (FC Bayern München).
Moussa Doumbia (FC Rostov) wins a free kick in the defensive half.
Substitution, FC Bayern München. Renato Sanches replaces Thiago Alcántara.
Foul by Arturo Vidal (FC Bayern München).
Timofei Kalachev (FC Rostov) wins a free kick in the defensive half.
Foul by Javi Martínez (FC Bayern München).
Aleksandr Erokhin (FC Rostov) wins a free kick in the attacking half.
Delay over. They are ready to continue.
Substitution, FC Bayern München. Franck Ribéry replaces Douglas Costa.
Substitution, FC Rostov. Andrei Prepelita replaces Sardar Azmoun.
Delay in match Soslan Dzhanaev (FC Rostov) because of an injury.
Attempt saved. Arturo Vidal (FC Bayern München) left footed shot from outside the box is saved in the top right corner. Assisted by Thiago Alcántara.
Goal!  FC Bayern München 4, FC Rostov 0. Joshua Kimmich (FC Bayern München) header from the centre of the box to the bottom right corner. Assisted by Juan Bernat following a corner.
Corner,  FC Bayern München. Conceded by Denis Terentjev.
Attempt missed. Robert Lewandowski (FC Bayern München) right footed shot from the centre of the box is too high. Assisted by Douglas Costa.
Bayern Munich began their bid for a sixth European Cup title by hammering Champions League debutants Rostov 5-0.