The second seed defeated the 54th-ranked Spaniard 4-6, 3-6, 6-1, 6-4, 7-5 in a thrilling match on Centre Court that lasted nearly three and a half hours.
Murray faces Poland’s Jerzy Janowicz in Friday’s semi-finals after the 24th seed beat compatriot Lukasz Kubot in straight sets.
“I came through an incredibly tough match. It could have gone the other way. I found a way to get through and that’s all you need,” said Murray, the US Open and Olympic champion.
“I started to play more solid and really took my time when I had the chance.
“Towards the end it was an unbelievable atmosphere. It was great to get through that one.”
With closest rivals Roger Federer, Rafael Nadal and Jo-Wilfried Tsonga already out of his half of the draw, the weight of expectation from British fans and media was all on the 26-year-old’s shoulders.
“I know how good these players are and it’s everyone else who keeps trying to say that they’re not,” he said of his opponents, calling Verdasco “incredibly dangerous”.
Fans inside the 15,000-seater court were on their feet as Murray came back from the brink.
Outside, the world number two was roared on by supporters packing the Aorangi Terrace, where fans waving British and Scottish flags watched on the big screen.
Murray had not dropped a set going into the quarter-final, the second time he has achieved the feat.
But he was severely tested by Verdasco, who had only beaten Murray once in their previous nine tour-level meetings: likewise a five-setter in the 2009 Australian Open fourth round.
Murray served his first double fault to hand Verdasco the first set.
With the crowd getting nervous, Murray started the second set in lacklustre fashion but turned it around to break for a 2-1 lead.
However, Verdasco broke back as Murray hit a string of unforced errors.
The left-hander broke again for 5-3 with a lucky backhand return that hit the net and dropped over.
In the next game, Murray had three break points but Verdasco recovered as Murray repeatedly chose the wrong shot, and the Spaniard took the set.
Exasperated, Murray screamed at himself: “What are you doing?” as he let off steam at the break.
Giving himself a good telling-off seemed to do the trick as the Scot broke at the first opportunity in the third set. Reinvigorated, he raced on to take the set 6-1, coming in from the baseline much more to win points.
In the fourth set, Murray broke for 4-3 as the momentum swung his way. Serving for the set, he sealed it with a forehand smash.
The fifth set decider then became a battle of who would blink first.
Neither player gave an inch until Murray broke when Verdasco shot long, giving the Scot a 6-5 lead.
Murray sent down an ace to give himself three match points and won it when Verdasco hit long again, sending waves of relief and elation around Centre Court.
The match was being watched from the royal box by former Manchester United manager Alex Ferguson, who saw several of his fellow Scot’s matches en route to his maiden Grand Slam triumph at the US Open last year.
Murray extended his grass court winning streak to 16 matches as he booked his place in his 13th Grand Slam semi-final.
Verdasco, 29, was on his best-ever Wimbledon run by reaching the last eight.