It was always going to be difficult for the second period to match the high-octane tempo of the first, and instead the decisive moments, both mad and magnificent, were compacted into the final few minutes.
With extra time looming, substitute Camavinga was dismissed for two yellow cards within 24 minutes of coming on. The first was for a foul on Jamal Musiala in the 78th minute and the second for taking the ball and delaying a Bayern free-kick following a foul on Kane.
The red card was to prove costly for him, and his team as the hosts made the most of their advantage.
Luis Diaz struck in the 89th minute to make it 5-4 to Bayern on aggregate, before Michael Olise put the icing on the cake in stoppage time.
With Guler also sent off in the aftermath, Real boss Alvaro Arbeloa admitted: “With the red card everything was over.
“It’s unbelievable to send off a player for this action. It’s not possible in a player like that, match like that. We feel all of this is a bit unfair.”
For Bayern, a semi-final against holders Paris St-Germain will follow and boss Kompany said: “The boys were mentally strong to recover from setbacks.
“The fans helped us as well. We stayed calm and always felt that our moment would come. The boys deserved this win.
“We showed absolute belief and will to fight our way back into the game.”
Gerrard, a 2005 Champions League winner with Liverpool, described it as “an incredible game of football, end-to-end, fascinating tactical approaches”.
“Extra time was looking likely, but one moment of madness, indiscipline and silliness has cost Real the chance of getting to the last four,” he added.
Former Liverpool defender Stephen Warnock told Match of the Day: “Both teams were brilliant going forward, but defensively both have been poor.
“If you want to win the Champions League, can you play like that? It is going to be fascinating to see how that pans out when Bayern face Paris St-Germain.”