A more veteran striker, Alexis Sanchez, made a surprise start for Inter Milan and scored in their 2-1 win over Salzburg.
At 1-1 Kane gave Bayern the lead in the 73rd minute from Jamal Musiala's pass and then returned the favour six minutes later when he set up the youngster to clinch matters.
The England captain now has 11 goals and six assists since arriving in the summer from Tottenham.
Munich had led from Kingsley Coman in the eighth minute but Galatasaray levelled from Mauro Icardi's Panenka penalty.
The home team had several big chances to take the lead but Sven Ulreich denied them in what could be his last game before Manuel Neuer's expected weekend return from more than 10 months on the sidelines with a leg fracture.
Bayern have now won 16 group stage games in a row, and are unbeaten in 37 games overall and 19 away from home at this stage - all competition records.
Sanchez scored in the 19th minute for the veteran forward's first goal since returning to the club in the off-season. Oscar Gloukh briefly levelled for Salzburg but Hakan Calhanoglu scored what proved to be Inter's winner from the penalty spot in the 64th minute.
Coach Simone Inzaghi surprisingly dropped Marcus Thuram — who has been a revelation this season — to the bench in favor of Sánchez.
It was a rare opportunity for the Chile forward, who had only started one of Inter's 10 previous matches this season, and he proved it was the right decision.
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