Harry Kane scored a goal in each half as Bayern Munich beat Lazio 3-0 on Tuesday to reach the Champions League quarter-finals 3-1 on aggregate.

Kane headed Bayern level in the tie in the 38th minute, cancelling out Lazio's lead from the first leg with his first Champions League knockout goal in five years.

Thomas Mueller doubled the hosts' lead with a clever header before half-time and Kane doubled up in the 66th minute, tapping in the rebound from a Leroy Sane shot to seal Bayern's progress to the next round.

"It was a perfect night for us," Kane told Amazon Prime.

"Everyone with energy, all around the pitch, the way we played and created chances and pressed without the ball -- it was a top, top performance."

Bayern kept their slim hopes of silverware this season alive while likely giving outgoing coach Thomas Tuchel a stay of execution until the quarter-finals in April.

"It's the most difficult trophy to win in Europe, probably the world," Tuchel, who won the Champions League with Chelsea in 2021, told reporters.

"But it doesn't get any easier or more difficult if you have a long contract or if you're leaving at the end of the season."

Kane recognised the manner of Tuchel's exit was "unusual" but said "nights like this can change the season. I'm proud of the boys -- we just have to keep this momentum up."

Mueller, a two-time Champions League winner, said: "We're going to make sure everyone enjoys this feeling, nevermind what is in the newspaper yesterday or tomorrow."

"We're very happy because we know the importance of progressing further today."

The loss snuffed out Lazio's chance of a first Champions League quarter-final appearance since 1999-2000.

Six-time European champions Bayern came into Tuesday's match staring down the barrel of a last-16 elimination for just the second time in the past 13 seasons.

Three straight losses including the 1-0 defeat in the first leg in Rome amounted to Bayern's worst run since 2015, costing Tuchel his job less than a year after arriving in Munich.

With Tuchel allowed to keep the seat warm until the summer, the coach promised to be more "ruthless".

On Tuesday, he benched big-money signing Kim Min-jae for Eric Dier, while moving Joshua Kimmich -- a fixture in midfield for Bayern and Germany -- to right-back.

Bayern pushed Lazio back early with the fit-again Sane and Jamal Musiala going close.

Immobile regret

Scorer of the first leg's only goal, Ciro Immobile had a golden chance to extend Lazio's lead after a Matthijs de Ligt mistake on 36 minutes but dragged his header just wide.

The miss would prove costly with Kane breaking through moments later, heading in a scuffed Raphael Guerreiro shot to even up the tie.

Mueller, another to see more game time after the announcement of Tuchel's impending departure, doubled Bayern's lead in first-half stoppage time, heading in a De Ligt shot in trademark poacher fashion.

Despite a prolific run in front of goal since arriving from Spurs, the German media joked the famously trophyless Kane had cursed Bayern, in danger of a first season without silverware since 2012.

The England captain made sure Bayern stayed in Europe though, turning in from close range after Sane forced a save from Lazio 'keeper Ivan Provedel, his 33rd goal in as many appearances this season.

With Lazio striker Immobile taken off with a knee injury, the visitors looked toothless in attack as Bayern gently controlled proceedings to take the sting out of the game.

Bayern kept their first clean sheet in eight games with captain Manuel Neuer equalling the Champions League record of 57 set by Real Madrid legend Iker Casillas.