"Before the game we spoke about how we can rely on him" Neuer told DAZN. "He's a phenomenon and we're proud that he's playing in the team."
Bayern had won their past 16 Champions League group stage matches in a row, but with scores locked at 0-0 with 10 minutes remaining, the run looked set to end.
Kane however had other ideas, heading in a Joshua Kimmich free-kick, to put his side in front.
The goal survived two offside checks but Kane was not done, scoring again with four minutes remaining, to seal the victory and bring his tally to a remarkable 19 in 15 in all competitions since joining from Tottenham in the summer.
"I'm enjoying it," Kane said of his blistering start to life in Germany.
"That's why they brought me here. As a striker, I love to score goals, love being on the pitch and most importantly love winning games".
Cedric Bakambu scored deep into injury time to give the visitors hope, but Bayern held on for a one-goal victory.
Galatasaray, who had fought valiantly at home, with Bayern's 3-1 win far less dominant than the scoreline suggested, again left empty handed, but can still qualify in the group with two games remaining.
"We expected a tough game after the away leg but I thought we played better," said Kane.
"We improved in the second half. Overall we deserved to win and I'm happy we qualified."
Bayern manager Thomas Tuchel said his side were "a little sloppy" and "inefficient" but said his team "perhaps felt the strain" of playing just a few days after their 4-0 thrashing of Borussia Dortmund.
Six-time winners Bayern have now gone 38 games unbeaten in the group stages, another Champions League record.
