It was a gruelling clash between the pair on Court 1 at the All England Club, and after four hours and 51 minutes - the longest match of the tournament so far - Muller came out on top with a 6-3 6-4 3-6 4-6 15-13 victory.
The first set went Muller's way as he was the only one to convert a break point within the first 34 minutes of the encounter.
It took four games for Nadal to create a break point in the second set, but Muller held firm before snatching his opponent's serve and putting himself in a commanding two-set lead.
Nadal had it all to do, and he responded with a battling display, which began in the third set when he crucially went a break up to give himself a 3-1 lead.
It took three set points, but he eventually got the job done, and even though he suffered a potential wobble early in the fourth set, he held off Muller and broke again to draw level after two hours and 36 minutes of play.
The deciding set was as tense as expected, with games going by way of serve and neither player creating a break point for the first nine games.
A pivotal moment for Nadal came when he saved two match points before delivering an ace to bring the set up to 5-5.
The two-time champion later saved another two match points after failing to break serve in the previous game, despite having four opportunities to do so.
The intensity grew as both players held, and the deadlock was eventually broken in the 28th game when 16th seed Muller sealed the match with a break of serve.