For the seventh time in his grand slam career, the Scot managed to pull out a victory from two sets behind.
The world number two had not dropped a set all tournament and Verdasco, only the Spanish number nine, was not expected to trouble him too much.
Murray played poorly for two sets but found his game when it really mattered and came out on top in a hugely tense fifth set to win 4-6 3-6 6-1 6-4 7-5 after three hours and 26 minutes.
In the last four Murray will meet giant Pole Jerzy Janowicz, who beat Lukasz Kubot in straight sets.
Murray credited Verdasco for making his life so difficult and admitted he had contributed to his own downfall in the second set.
The 26-year-old said: "He played extremely well in the first set and in the second set I made some poor choices and he served fantastically well.
"He is a very, very good player, he used to be at the top of the game.
"I did hold my nerve at the end of the match, didn't make any bad choices like I did in the second. I played more solid and took my time when I had the chance.
"It's not a warning because I know how good these players are, it's everyone else who is saying that they aren't. I can lose these matches if I don't play my best."
Sir Alex Ferguson was watching from the Royal Box, and when it was put to Murray that he may have benefited from a dose of the famous hairdryer treatment after the second set, he bristled and said on the BBC: "I don't know it all, far from it. But I don't see why I should get told off after that.
"I tried incredibly hard, chased every single ball down from the first to the last and I came through an incredibly tough match. It could have gone the other way but I found a way through."
Verdasco was the first left-hander Murray had faced this season but he generally likes playing them and had beaten the Spaniard in eight of nine previous meetings.
Despite Verdasco suffering a career dip over the last two years to be only ranked 54th, his big serve and forehand made him a potential danger.
Watching from the Royal Box was former Manchester United manager Sir Alex Ferguson, who was at courtside when Murray won the US Open last September.
There was no real rhythm in the early stages. Murray had chances but the Scot's second serve was a concern.
He got out of trouble in the eighth game but Verdasco took advantage again at 5-4 and, in an effort to save a set point, Murray double-faulted.
The crowd attempted to gee up their man, and he could breathe a little easier when Verdasco missed a forehand to offer up a first break for 2-1 in the second set.
But Murray handed it back three games later with two poor points and was then broken again to trail 5-3, some extremely tight play from the Scot exacerbated by a lucky net cord for his opponent on break point.
It was crisis time, but Murray usually relishes a scrap. He showed his best defensive skills as Verdasco threw everything he had at him only for the Scot to win a titanic point with a scooped forehand pass.
The crowd urged their man on during the point, and Ferguson rose to his feet in delight at the winner.
Murray had three chances to break back, but Verdasco saved them and pounced on his first set point when the second seed sent a backhand flying over the baseline.
With his head slung low, Murray returned to his chair and shouted at himself.
But he made the perfect start to the third set with a break in the second game and pulled away.
Nervousness gripped the crowd again when Verdasco brought up a break point in the second game of the fourth set.
Murray's serve came to his rescue to save that one and a second that followed, and the Scot held after a crazy point that ended with both men on the grass.
The second seed was clinging on as he saved two more break points in the sixth game with big serves, and he got his reward as he broke for 4-3.
Murray threaded a backhand down the line to bring up break point and this time Verdasco overcooked a forehand.
It was advantage Murray but he knew from experience the job was not yet done. In the US Open final he lost the third and fourth sets from two sets up but won the match, beating Novak Djokovic.
Verdasco began confidently, and it was easy to understand Murray's frustration when the Spaniard hammered down a second serve ace at 3-3 and 15-30.
Murray swiped at the grass in frustration as another ace gave his opponent the game, and there was a sense the next game was a very big one indeed.
Suddenly it was 0-30, but four points in a row gave the Scot the game and the crowd roared.
Both players knew one opportunity could settle the issue, and it went Murray's way.
In the 11th game, the Scot won a titanic rally to bring up the first break point of the set and took it, forcing Verdasco to hit long.
A chant of "Andy, Andy" went around Centre Court, and he made no mistake serving it out.
Murray brought up three match points, clinching the first when Verdasco hit a backhand long.
The Spaniard challenged but confirmation it had missed brought relief for Murray, the whole of Centre Court and beyond.