Murray showed incredible fight as he came up against a fierce competitor in Tiafoe, who was in control for much of the match.
"I think that's the first time in my career I've played a 7-6, 6-7, 7-6. I don't think I've ever played a match like that," Murray said.
"I think it's the longest three-set match I've played by a distance. I'm tired, obviously.
"It was an unbelievable battle. He kept coming up with brilliant serves and great shots when I was getting chances.
"I don't mind playing long matches but that was taking it to another level."
The American served for the opening set twice and needed six set points to win the second set, led by a break in the decider and also held two match points in the final-set tie-break.
Tiafoe however was unable to take advantage on two match points however and it let Murray in. Murray showed his power as he struck 21 aces to stay in contact with his opponent and saved seven of the 10 break points he faced.
He has now set up a second-round clash against second seed Diego Schwartzman for a place in the quarter-finals held in Antwerp.