Kylian Mbappe's strike 16 minutes from time gave record 15-time winners Real a lifeline at their Bernabeu home in a gripping battle.

Vincent Kompany's side, arguably the strongest team in Europe this season, produced a masterclass in the first half and could have been several goals ahead.

Instead they had only one from Diaz, which Kane added to after the interval, but French superstar Mbappe's goal kept Alvaro Arbeloa's side in the tie.

"It's a shame it wasn't 2-0 at the end to be completely honest," said Bayern goalkeeper Manuel Neuer, who made several good saves in the second half as Madrid ramped up the pressure.

"You know how dangerous Real Madrid are, we saw that again today. They had plenty of chances and could've scored more.

"First up we're happy about the victory here away from home, but it'll be a tough challenge back in Munich."

Real Madrid defender Antonio Rudiger said that Mbappe's goal had kept the tie open.

"For me, the two goals we conceded were gifts, and in the second half we needed more," Rudiger told Movistar.

"I think we're alive, everything is open with that goal from Mbappe. I think we had many chances to score goals, but in the end this is how it is."

The round's glamour fixture was a European classic -- a record 29th time these heavyweights have faced each other in the competition.

Bayern flew out of the blocks and Konrad Laimer's rasping effort narrowly over the bar showed that Kompany's side had come to the Spanish capital to attack.

After nine consecutive matches against Madrid without a win, Bayern were determined to put an end to that streak and show why many consider them favourites to lift the trophy this season.

Bayern should have taken the lead through defender Dayot Upamecano, who fluffed his lines from point-blank range, after Kane cushioned the ball into his path, allowing Alvaro Carreras to hack the ball off the line.

Madrid were largely pinned back as Bayern dominated the ball, but the hosts sought to send through rapid strikers Mbappe and Vinicius Junior on the counter, with Neuer saving from both.

Madrid's defensive work was nearly undone by a moment of madness from 18-year-old midfielder Thiago Pitarch, starting in place of Jude Bellingham, whose poor back pass under pressure was straight to Serge Gnabry in front of goal, but goalkeeper Andriy Lunin deflected the German's shot behind.

- Slick Bayern -

Bayern toyed at times with Real, who ceded possession far too often, and their opening goal was a slick combination play which caught the hosts on their heels.

Gnabry exchanged a one-two with Kane and pushed the ball in behind Trent Alexander-Arnold for the Madrid defender's former Liverpool team-mate Diaz to reach and stroke home past Lunin in the 41st minute.

Kane, who was a doubt to start because of an ankle problem, did not seem fully fit but made an even bigger impact less than a minute into the second half.

Olise cut the ball back to him on the edge of the box and Kane fizzed a curling effort into the bottom corner to double Bayern's lead.

It was his 11th Champions League strike this season, trailing only Mbappe on 14, and his 49th goal across all competitions.

Vinicius had a fine chance to pull one back for Real when Upamecano badly underhit a header back to Neuer, but the Brazilian sliced his shot into the side-netting.

Neuer made a brilliant save to thwart Mbappe once more and left with the man of the match trophy.

With Bellingham on as a substitute, Madrid stepped up a level and eventually found the net with Alexander-Arnold guiding a low cross to the far post where Mbappe beat Neuer, smashing home off the crossbar after 74 minutes.

Bayern angrily roared back with Kane volleying wide as they tried to restore their advantage, but instead they take a slender lead back to Bavaria for next Wednesday's second leg.