Murray missed four match points in the second set but ensured there was no miracle comeback for Robert by wrapping up a 6-1 6-2 6-7 (6/8) 6-2 victory on his sixth chance.
Much greater tests await with either Roger Federer or Jo-Wilfried Tsonga waiting in the last eight.
Murray raced through the first set of his fourth-round clash with France's Stephane Robert at the Australian Open.
The world number four needed just 26 minutes to wrap it up 6-1, ending with a run of 11 straight points.
Of all the possible opponents Murray could have had in the fourth round, Robert was among the most unlikely.
The 119th-ranked Frenchman initially lost in the final round of qualifying but secured a lucky loser spot 10 minutes before meeting Aljaz Bedene after Philipp Kohlschreiber pulled out.
Robert won that one and two more to become the first lucky loser ever to reach the fourth round at Melbourne Park.
The 33-year-old is an engaging character and showed a willingness to go for his shots in the first game that earned him a break point.
But Murray saved it and moved into a 3-1 lead when Robert overcooked a forehand.
The Frenchman blew another chance in the fifth game, and Murray was not nearly so charitable, stringing four good points together to move 5-1 ahead and then clinching the set with ease.
Murray had won 23 straight points at the end of his second-round win over another Frenchman, Vincent Millot, but 11 was as far as he got this time before Robert won the first point of the second set.
That did not stop him losing his serve, though, the drop shot proving as erratic for Robert as it sometimes is for his opponent.
Robert had never previously been beyond the second round of a grand slam, with his greatest moment prior to this tournament coming when he beat Tomas Berdych at the French Open.
The veteran was beginning to look rather out of his depth here as Murray eased into a 4-0 lead.
Robert finally stopped the rot after nine games in a row to Murray, and that spelled the end of the Scot's purple patch - not that he was ever in trouble.
He double-faulted on his first two set points but took it 6-2 with a forehand winner on the third chance.
It was a much more even finish to the set, and that continued in the third, with Robert holding his serve comfortably for the first time in the match.
Murray was struggling to stamp his authority on points in the way he had been earlier in the match, and he saw six break points slip away in the seventh game.
Frustration was etched on the Scot's face as he sought the decisive breakthrough but on his seventh chance he finally made it, threading a backhand return winner down the line.
Robert, who won only four games against British number three James Ward in Brisbane earlier this month, was fighting until the last and giving a good account of himself.
He created a fourth break point of the match to give himself a chance to level at 4-4 but it went the way of the other three and Murray held on.
It was certainly not proving easy for the British number one to get over the line, though, and, after Murray missed two match points - one with a double fault - Robert secured his first break of the match.
The Frenchman deserved his reprieve, and he made a flying start to the tie-break by winning two points on Murray's serve to lead 3-0.
The Scot pegged him back and had two more match points but a poor second serve and a missed forehand accounted for those, too.
Murray then netted a forehand to give Robert his first set point, and a forehand long handed it to the Frenchman 8-6, Murray taking his frustration out on his racquet.
Murray's first dropped set of the tournament appeared more of an inconvenience than anything more threatening, but he still needed a good start to the fourth.
And he got it with a break to love to lead 3-1, Robert rather handing it to him by double-faulting for the sixth time.
The 33-year-old had already called the trainer to have his thigh massaged, and it was an awfully long way back for him.
Murray thought he had taken his fifth match point when Robert's shot was called wide but it was correctly overruled.
A sixth arrived straight after, though, and this time he clinched it when the Frenchman netted a forehand.
It took two hours and 21 minutes but Murray was finally through to his 12th grand slam quarter-final in a row at tournaments he has contested.
Robert stayed on court to soak up some deserved applause as a relieved Murray celebrated surviving his first hiccup.