Cristiano Ronaldo's two away goals came after Arturo Vidal had opened the scoring for the hosts, but the midfielder then spectacularly missed a first-half penalty to make the score 2-0.
Here's what we learned from an entertaining encounter...
Neuer reminds us he is the new master
On Tuesday night, we saw arguably the save of the season from a man who was widely considered the best goalkeeper in the world until a few years ago. Here, Gianluigi Buffon’s successor bested him.
The first replays appeared to suggest that Manuel Neuer had not reached Karim Benzema’s header. It looked as if he was fortunate for it to bounce down off the crossbar and fall just in front of him.
Yet another look revealed that Neuer had ever so slightly connected with the ball and done so with enough force to divert its course, diving at a quite an extraordinary height in the process.
It was as brilliant, maybe better, than Buffon’s stop 24 hours earlier, and a reminder that goalkeeping now has a new master.
Ronaldo ends his drought at the perfect moment
Arturo Vidal’s penalty miss on the stroke of half-time never seemed as though it would be enough to shake Bayern Munich on its own. Madrid needed to capitalise on the Chilean’s horrendous miss and had to do so as soon as possible.
What better time, then, for Cristiano Ronaldo to end his long Champions League goal drought than less than two minutes after the re-start?
His last strike in this competition had come in September, but this one came at the perfect moment, compounding Vidal’s miss and levelling a tie that could so easily have got away from Madrid.
A ‘clutch’ player, delivering at a ‘clutch’ moment. His second goal won this leg, his first turned it.
Bale spends his night on the peripheries
The rabid, often unfair criticism that once followed Gareth Bale’s Real Madrid career has subsided as his performances have improved, but here he suffered a night to forget.
The Welshman struggled to impose himself on a game that, while he was on the field, the hosts dominated.
The one moment where he looked like making a significant impact came in the 56th minute, as he rose to meet Luka Modric’s cross and blast a header at goal. Neuer denied him.
Minutes later, Bale was leaving the pitch, replaced by a talented but inexperienced 21-year-old. He will be disappointed with his evening.
Premier League teams compare poorly to these two
Like with the few other members of the Champions League’s shrinking elite, it is easy to forget how good these two teams are when you watch them play each other.
The intent behind each side’s build-up and the control in their play are both easily missed when two teams are so evenly matched.
To fully appreciate their quality, you simply have to wonder how any Premier League club – Chelsea apart, perhaps – would handle them. They would be ripped to bits.
Of course, in the case of Arsenal, we have a recent and telling body of evidence. Yet the same Bayern team that ripped Arsene Wenger’s side apart was contained for periods here.
It’s a tired point, but this game was a reminder of why the members of European football’s moneyed elite are simply streets ahead of their chaotic English peers.