The sixth seed from Luxembourg prevailed over Barton 6-1, 3-6, 6-4 as temperatures nudged 42 Celsius.
The sixth seed from Luxembourg prevailed over Barton 6-1, 3-6, 6-4 as temperatures nudged 42 Celsius (108 Fahrenheit) to reach the quarter-finals.
Muller, who lost to Grigor Dimitrov in last year's semi-final, will next face Uruguayan second seed Pablo Cuevas, who overcame Frenchman Nicolas Mahut 6-4, 2-6, 6-2.
"It was tough today. I think it was good that we both have big serves, there were not many rallies so that helped to get through it. I was happy when it was over," Muller said.
The 33-year-old added that he was still struggling with the lingering effects of illness from last week's Brisbane International, where he lost in the first round to American Jared Donaldson.
"The result is actually not so bad, the quarter-finals so I should be happy," he said.
"I like it here. The courts are quite fast, I enjoy the city. I feel good here. I don't know what it is, but it's true my results have been great here over the last three years."
German fifth seed Philipp Kohlschreiber knocked out Australian Jordan Thompson 7-5, 6-4 and will face the winner of the match between two-time defending champion Viktor Troicki and Paolo Lorenzi.
Kohlschreiber needed just 84 minutes to cruise past Thompson, who failed to save a single break point.
Spanish fourth seed Pablo Carreno-Busta dropped the first set before overcoming German Mischa Zverev 3-6, 7-6 (7/4), 6-3
Tournament top seed Dominic Thiem plays his first match of the Australian Open warm-up event in an evening clash against Portuguese qualifier Gastao Elias.