Frenchman Pouille, the 28th seed, had lost in the first round on his five previous visits to Melbourne but recovered from a break down in the opening set to win 7-6 (4) 6-3 6-7 (2) 6-4 and set up a match with Novak Djokovic.
Pouille, ranked 31, reached the quarter-finals of Wimbledon and the US Open in 2016 but struggled to build on that and endured a disappointing 2018.
There was little to indicate he was about to go on such a run after beginning this season with four defeats having persuaded Mauresmo to give up her new role as Davis Cup captain to coach him.
The 24-year-old said: “She has the right mindset, she knows everything about tennis. It’s not about being a woman or a man, you just have to know what you’re doing and she does.”
Six-time champion Djokovic benefited from the retirement of Kei Nishikori with the eighth seed trailing 6-1 4-1.
Nishikori had come through three five-set matches, including a five-hour epic against Pablo Carreno Busta. He received treatment to his right thigh at the end of the first set before deciding he could no longer continue.
Jamie Murray suffered double quarter-final disappointment.
The Scot and Brazilian partner Bruno Soares were beaten 6-3 6-4 in the men’s doubles by 2017 champions Henri Kontinen and John Peers, then Murray and Bethanie Mattek-Sands lost in the last eight of the mixed doubles 6-2 7-6 (5) to Australian pair Astra Sharma and John-Patrick Smith.