Mueller, an 86th minute substitute for Robert Lewandowski, received the ball just inside the Arsenal penalty area, shaped to shoot with his right foot but transferred it to his left and thrashed past goalkeeper David Ospina. His joy was unalloyed. Sanchez slumped to his haunches near the touchline.
The second leg in the Uefa Champions League tie takes place on 7 March. Only extraterrestrial supporters would bet on Arsenal's progress to the last eight.
Arsenal manager Arsene Wenger admitted the north Londoners had collapsed mentally in the second half.
"We conceded the second goal and then the most important thing was that we lost Laurent Koscielny to injury," said Wenger. "I am not looking for excuses. Overall I must say they were a better team than us, they played very well in the second half and we dropped our level."
Wenger, 67, who faces the prospect of a seventh successive last 16 exit from European club football's most prestigious competition, added: "Their third goal was a killer - we had no response. It is a shock of course to lose at this level."
Arjen Robben opened the scoring for Bayern after 11 minutes. Sanchez levelled on the half-hour, hitting in the rebound after Manuel Neuer saved his penalty.
But Robert Lewandowski restored the German side's lead after 53 minutes, heading in a cross from Philipp Lahm with Thiago Alcantara adding a third just three minutes later.
Thiago hit the fourth just after the hour following sustained Bayern pressure before Mueller guilded the evening's annihilation.
Bayern skipper Philipp Lahm said: "Before Arsenal's penalty we were very good but after that and up until half-time we weren't really good. But in the second half we were sensational."