Murray was just two games from defeat before his 20-year-old opponent crumbled with the finish line in sight. As he so often does, Murray pounced and secured a place in the last 16.
The Scot has one of the best records on tour against left-handed players. Excluding his matches against the world No.1 Rafael Nadal, he held a 48-3 career-record against lefties going into the tie, but win No.49 was perhaps his ugliest.
He will have to improve significantly if he is to beat Milos Raonic or Alejandro Falla the last 16 tomorrow and stay in line for a potential quarter-final meeting with Nadal.
There were no signs early on that the match would turn out to be anything like the crazy spectacle that was to unfold. Murray raced to a 3-0 lead with a double break as the world No.77 Vesely, playing on the biggest stage he has graced, made a nervy start. Murray, though, failed to take control of the match from there and gifted one of the breaks back, playing a sloppy service game as the Czech began to find his feet.
Murray suffered a severe lapse of concentration when serving for the set at 5-4 and Vesely levelled after a correction on a line call by the umpire Mohamed Lahyani had been overturned by Hawk-eye.
The tiebreak was one of the poorest Murray has played in recent years, although his cause was not helped as Lahyani made another error as he failed to spot that Vesely had reached over the net to play a volley.
The match went from bad to worse for Murray as he wildly overhit a couple of balls in the opening game of the second set to gift Vesely a break. The Scot claimed it back as Vesely began to show signs of edginess, missing three simple overhead smashes and Murray piled the pressure on Vesely who was serving to stay in the set. The young Czech saved a couple of set points but was beaten by a drop shot on the third.
Going into the deciding set, it seemed reasonable to anticipate that the two-time grand slam tournament winner Murray would take command of the match. Instead, he continued to play passively, was broken in the opening games and, after swopping a coupe of breaks, trailed 4-2.
Once again, though, the scent of a real upset proved too much for Vesely. His tenth double fault enabled Murray to draw level at 4-4 and the Scot made him pay, finally securing victory on his third match point.