The England skipper said there would be no panic though and predicted that the two sides could well clash again at the business end of the tournament next spring.
"It was a tough game which is kind of what we expected. It was a good battle in the first half which was fairly even," Kane, who has bagged 29 goals in all competitions for club and country so far this season but was quiet on Wednesday, said.
"Second half we didn't quite have the same energy or intensity and we lost too many duels. It's our first loss of the season. We don't want to panic too much about it. But we will learn about it for sure.
"I'm sure we'll see them (Arsenal) again in the later stages of the Champions League."
During his record-breaking career at Tottenham Hotspur, Kane bagged 14 goals in 17 North London derbies in the Premier League. He also scored a penalty for Bayern in the first leg of the Champions League quarter-final against Arsenal two seasons ago when the Bundesliga side won 3-2 on aggregate.
But Arsenal have grown in stature since then with a huge summer spend giving them a squad depth that not even Bayern can match. Arsenal were missing key defender Gabriel, 63.5 million pounds ($84 million) striker Viktor Gyokeres and others on Wednesday.
However, manager Mikel Arteta still had a bench packed with match-winners and it was substitutes Noni Madueke and Gabriel Martinelli whose second-half goals sealed a fifth win from five games in this year's competition to send Arsenal top.
"We are different, they are different as well. It was a different stage of the competition and it was thin margins how we went out," Arteta said when asked what the difference is between his side now and the one from two years ago.
"We learn, we are better."
Arsenal top the Premier League by six points and are now three points clear at the top of the Champions League group and have one foot already in the last 16.
However, man-of-the-match Declan Rice said there was no chance of Arsenal getting complacent.
"That's just a lot of talk. The Champions League - you're either in or you're out," he said. "At the moment, it's only the league phase, anything can happen. Look at last season, Liverpool finished top, then they faced (Paris St Germain) and they were out.
"We just have to take it one game at a time, stay focused and be ready for whatever is thrown at us, because on our day I think we can beat anyone."
($1 = 0.7552 pounds)