Cilic, who triumphed 3-6 7-6(6) 7-5 5-7 6-1 will now face Sam Querrey in the semi-finals after the American upset home favourite Andy Murray.
The 34-year-old Muller was an unexpected presence in the last eight after he pulled off a surprise, five-set win over Rafa Nadal in the fourth round.
A day off ensured decent recovery time from the nearly five-hour battle with Nadal, which included a fifth set that lasted over two hours.
And Muller, the journeyman who had not even won a regular ATP Tour event until this year, succeeded in carrying his inspired form against the Spaniard into this last eight contest.
Muller broke in the seventh game with a powerful drive down the line, shouting out his delight as he clearly sensed another upset could be on the agenda.
He broke again to win the set when Cilic went wide with a return and his small but noisy contingent of supporters from his Luxembourg homeland roared their delight.
Cilic was not rattled though and he went into the second set tie-break looking strong, taking advantage of an ill-timed double-fault from Miller at 6-6, securing set point with a fantastic cross-court winner.
The third set was another tight affair and again it was an unforced error from Muller which gave Cilic the upper hand. Serving to stay in the set, Muller was broken to love, the final point a frustrating volley into the net.
The clever serve and volley play and the ice-cool demeanour that put paid to Nadal’s hopes, had gone but Muller was not ready to leave for his home in Leudelange just yet.
Cilic’s serve was firing dangerously but Muller held on and then struck to go 6-5 up, breaking with a fine cross-court winner and uncharacteristically dancing across to his chair.
He regained his composure quickly and calmly served out for the set but his revival was short-lived.
Cilic broke Muller’s first service in the deciding set and the resistance crumbled, the Croat charging through the games to win the final set 6-1.
After three straight defeats in the last eight at the All England Club, Cilic tastes the semi-finals for the first time and will surely sense the chance to go even further.
(Reporting by Simon Evans; Editing by Alison Williams)