The Bundesliga champions led 2-0 at the interval, with Leroy Sane and Serge Gnabry on the scoresheet, before Rasmus Hojlund notched his first Man United goal early in the second period.
Bayern restored their two-goal advantage four minutes later from the penalty spot, though, with Kane dispatching after Christian Eriksen had been penalised for handball.
Casemiro's late effort set up a grandstand finish in Munich, but Bayern netted again through Mathys Tel to all but make sure of all three points.
There was time for another goal, with Casemiro heading a Bruno Fernandes free kick into the bottom corner, only for the referee's whistle to blow almost immediately after the restart.
Man United have now lost their last three matches in all competitions, while Bayern have taken an early control of Group A, sitting at the top of the section after the first set of fixtures.
The visitors arrived in Munich without a whole host of key players due to injury and personal issues and indeed off the back of a 3-1 defeat to Brighton & Hove Albion in the Premier League.
Facundo Pellistri came into the away side's XI for Scott McTominay, while there was a certain Kane at the tip of the Bayern attack, with the England captain bidding to add to the four goals that he had already registered for the Bundesliga champions.
Sane managed to break beyond the Man United defence inside the first three minutes, but the Germany international's effort from a wide area was deflected wide of the post by Lisandro Martinez.
Man United had a golden chance to take the lead in the fourth minute when Christian Eriksen, who had broken into the Bayern box, delivered a low cross into Pellistri, but Alphonso Davies made a wonderful last-ditch challenge before Eriksen was denied by Sven Ulreich.
Bayern's goalkeeper then had to deny Marcus Rashford in the 18th minute, with the visitors starting the better at Allianz Arena.
The home side had a half-chance in the 20th minute when Jamal Musiala was released by Kane, but Victor Lindelof was across to block the Germany international's strike.
Martinez then defended brilliantly against Gnabry two minutes later, with the Bayern attacker threatening to break into a one-on-one situation with Red Devils goalkeeper Andre Onana.
Bayern managed to make the breakthrough in the 28th minute when Sane's effort from distance somehow squirmed through the arms of Onana in a terrible mistake from the Red Devils goalkeeper.
The hosts then doubled their lead in the 32nd minute, with Musiala driving down the left before delivering into Gnabry, who was unmarked and comfortably picked out the bottom corner.
There were no takers on a dangerous cross from Reguilon in the 35th minute, before Fernandes missed the target from distance and Hojlund guided a header wide, with the visitors looking to respond, but the atmosphere had completely changed inside the stadium with the quick-fire double.
Sane bent one just wide of the far post late in the first period, with Bayern threatening to completely take the match away from their opponents just before the half-time break.
Man United were back in the match in the 49th minute, though, with Hojlund's effort from inside the Bayern box managing to find a way into the back of the net off the foot of Kim Min-jae.
Musiala had one deflected just wide of the post down the other end moments later, before VAR sent the referee across to the monitor to look at a handball from Eriksen inside the box, and a penalty was awarded for Bayern in a hammer blow to the Red Devils.
Kane, as expected, stepped forward to find the back of the net for the Bundesliga champions, picking out the bottom corner with a clinical strike.
Sane hit the post in the 56th minute following more excellent work from Musiala, who was the best player on the pitch at Allianz Arena.
Sergio Reguilon flashed one across the face of the Bayern goal on the hour, with Man United looking dangerous on the transition, but they just could not seem to keep the door shut down the other end.
Onana did well to keep out Kane and then Sane during a strong period for the home side, who came close again in the 78th minute, with Eric Maxim Choupo-Moting being denied by the Man United goalkeeper.
Kingsley Coman was the next Bayern player to be denied by Onana, who recovered well from his mistake to put in a solid-enough performance in Munich.
Alejandro Garnacho sent a Rashford cross wide of the post late on, with Man United looking for a second which would have set up a grandstand finish, and Casemiro got his name on the scoresheet in the 88th minute when he somehow squeezed the ball into the bottom corner after falling to the ground.
Thomas Muller turned one against the Man United post late on, but Bayern made sure of the win in stoppage time when Tel expertly found the back of the net from inside the box.
There was time for another goal, with Casemiro heading a Fernandes free kick into the back of the net to complete his brace, but the referee's whistle blew shortly after the restart.
Man United will be bidding to avoid a fourth successive loss in all competitions when they head to Burnley in the Premier League on Saturday night, while Bayern will be back in Bundesliga action at home to Bochum earlier in the day.