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.
Murray said of Robert: "He's a fun player to watch but not fun to play. He plays all the shots, he's very unorthodox. He made it very tough for me.
"If you can, you want to close matches out when you have the opportunity. I had chances on my serve, I didn't get them. I tried to stay focused at the beginning of the fourth, got an early break and that helped."
Murray was coasting at two sets up but squandered four match points in the third, which Robert won on a tie-break. The Scot took his frustration out by breaking his racquet.