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There were a few nerves at the end but Liverpool set up a Champions League semi-final with Villarreal after Roberto Firmino scored twice in a 3-3 second-leg draw with Benfica on Wednesday to ensure a 6-4 aggregate win.
Six-time European champions Bayern, who had won 12 of their previous 13 home games in the competition, levelled the tie when Robert Lewandowski drilled the ball in off the post after 52 minutes with his 13th Champions League goal of the season.
Villarreal took the game to Bayern and should probably have won by a larger margin given their clear opportunities.
Having fallen before the quarterfinals against Olympique Lyonnais and Porto in the previous two seasons, in-form Juventus, who had lost just once in their last 21 matches in all competitions, were firm favourites to progress.
Holders Chelsea will face French champions Lille, just as they were drawn in the original draw, while six-times champions Bayern Munich will now face RB Salzburg.
Rangnick steps in to replace Solskjaer, who was removed after United's 4-1 defeat by Watford this month.
Chelsea claimed a thumping 4-0 victory over Juventus.
Former player Michael Carrick, a member of Solskjaer's coaching staff, will take charge of the United team for Tuesday's Champions League match away to Villarreal.
Manchester United suffered a fourth defeat in five Premier League games as they went down 4-1 at Watford to pile more pressure on manager Ole Gunnar Solskjaer.
The 36-year-old Ronaldo has now scored five goals in four games for United in this Champions League campaign and has nine in 11 outings for the club this season.