In their first ever meeting, Australian Open champion Wawrinka broke his opponent from Luxembourg in the opening game and pocketed the first set of the quarter-final quickly.
Mueller, however, refused to throw in the towel and took a 4-2 lead in the second before Wawrinka rallied to level it 4-4 and force a tie-breaker in which the Swiss world number four prevailed.
“Gilles is a big-server, left, it’s tough returning but I’m happy,” Wawrinka, who meets fourth seed David Goffin in the semi-final, said in a courtside interview.
“I was really aggressive and I’m happy the way I finished the match, the way I’m playing in general,” said Wawrinka who hit 29 winners, six more than his opponent.
Spaniard Roberto Bautista will meet Slovenia’s Aljaz Bedene in Saturday’s other semi-final.