The Scot, seeded No.4, took the opening two sets by 6-2, then showed all his battling qualities to battle back from 5-1 - winning 24 out of 25 points - down in the third to win 7-5.
His next opponent will be Spain's Feliciano Lopez.
The roof over Rod Laver Arena, which had been closed first for heat and then rain, opened as Murray and Millot warmed up.
It was humid and still very warm at 10pm local time, but nothing like the conditions that prompted the tournament to implement its extreme heat policy during the afternoon.
Millot, a qualifier ranked 267, was playing by far the biggest match of his career and he quickly looked out of his depth.
Murray broke in the opening game and then again to lead 3-0 despite not having hit a single winner.
Millot's play in the fourth game suggested he may have been affected by nerves to start with as he took advantage of two double faults from Murray to break serve.
The Scot looked less than impressed and went on the attack, breaking again with a precision backhand winner down the line.
Murray had a set point on the Millot serve but did not take it, and then saw three more go begging, one with his fourth double fault of the match.
A let court robbed him of an ace on his fifth chance but he dismissively placed the ball in the other corner moments later.
Murray, who had played very well in his first-round demolition of Go Soeda, did not appear to be enjoying the experience very much while Millot was growing into the match.
But there was a huge gulf in class between them and it was just a matter of when that told.
Millot was going for broke and it cost him in the third game of the second set, while Murray broke again to lead 4-1.
Millot, who once harboured hopes of following his father into professional football, held for 5-2 but then took a medical time-out to have his calf rubbed.
The 27-year-old played for more than three hours in beating Wayne Odesnik in the worst of the heat on Tuesday for his first grand slam victory.
After the stoppage, Murray came out to serve for the set, and he won the game to love to take it 6-2 before that dramatic final-set comeback.