Kane headed in Michael Olise's free kick from the right in the 61st minute, taking his Bayern tally in the competition to 34 goals in 39 appearances. His current run is one match short of Robert Lewandowski's nine and three behind Cristiano Ronaldo's record of eleven.
Bayern had to wait until after the interval to break through. Jamal Musiala scored in the 47th minute following a contribution from Kane. Alphonso Davies later struck from distance, while Olise finished with two goals and an assist.
Goalkeeper Nikita Haikin had made five first-half saves before injury forced him off. Bayern became more effective after the break and secured a convincing opening win. Kane's next opportunity to extend his European run is scheduled against Viking on October 13; matching the next milestone remains a future possibility.