For the seventh time in his grand slam career, the Scot managed to pull out a victory from two sets behind.
The world No 2 had not dropped a set all tournament and Verdasco, only the Spanish No 9, 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 tense fifth set to win 4-6 3-6 6-1 6-4 7-5 after three hours and 26 minutes.
Murray will now meet giant Pole Jerzy Janowicz, who beat Lukasz Kubot in straight sets yesterday.
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 Scot 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.”
“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.”
Janowicz saw off his fellow countryman and close friend Lukasz Kubot to become the youngest Wimbledon semi-finalist since his next opponent, Murray, four years ago.
The imposing 22-year-old, standing at 6ft 8in, fired 30 aces and wrapped up a 7-5 6-4 6-4 victory over world No 130 Kubot in just over two hours.
The win sees him becomes Poland’s first ever man to reach a grand slam singles semi-final and the occasion clearly meant a lot to both players, who embraced and swapped shirts at the end of the match.
“I am just really, really happy,” a clearly overwhelmed Janowicz said. “I didn’t expect to be able to go that far in a grand slam, in Wimbledon.
“I am just very happy. I have not many words to say right now. I am just really happy and it doesn’t matter if I am the first or second [Pole in the semi-finals]. This is my first semi-final and I have nothing to say, basically.”
In the other side of the draw, Juan Martin del Potro had “magic pills” to thank as he also reached the semi-finals for the first time to set up a clash with world No 1 Novak Djokovic.
The Argentinian eighth seed suffered a nasty fall in the opening game of his match with fellow wounded warrior David Ferrer, but recovered to post a 6-2 6-4 7-6 (7/5) victory on Centre Court.
Del Potro admitted he was “really close” to having to pull out of the match, but his bravery paid off.
“I think it’s going to be dangerous if I’m not careful in the next few days,” he said after coming off court.
“I think I played my best tennis of the tournament against David and I’m so happy to be in the semis.”
The 2009 US Open champion sustained the initial setback when beating Grega Zemlja in the third round, but put up with it and battled through a last-16 tussle with Andreas Seppi to set up yesterday’s contest with the Spanish fourth seed, who has been suffering with a sore big toe.
“[I was] really close [to quitting] because I felt a lot of pain. It was exactly what I did before,” Del Potro said.
“It’s really painful, I twisted my knee once again. The doctor gave me some magic pills.”
Defending champion Djokovic came through his toughest test so far, beating the hard-hitting Tomas Berdych in straight sets to make his 13th consecutive grand slam semi-final.
Djokovic wasted four break points in the first set and was broken twice in the second, but he ultimately came through to record a deserved 7-6 (7/5) 6-4 6-3 win against the seventh seed.
The top seed, who is yet to drop a set in the tournament, will play Del Potro in the last four and is still on course to play Murray in the final.
Djokovic admitted afterwards that he thought he could have lost the match.
He said: “It was a strong start for me today.
“I had a few break points, a few chances to break in the first set, but I didn’t manage to do that and the tiebreak was decided by one mistake with a forehand at 6-5. It was a very close match, it could have gone either way.
“He could have won the first two sets, he had a double break in the second.
“I don’t know how I managed to go ahead, I don’t know how I turned it around. I am really happy with the performance. I am playing some of the best tennis on grass of my career.”
Djokovic was glad to see fellow Serb and Manchester United captain Nemanja Vidic in the crowd, along with former Chelsea striker Andriy Shevchenko, who had also come to watch.
“It really means a lot to see Nemanja with Andriy here,” Djokovic said.
“I have only met [Vidic] once before briefly, but we are glad that he accepted the invitation to come.”
Today is the turn of the women and will see the two semi-finals take place on Centre Court.
And though Serena Williams’ conqueror Sabine Lisicki has emerged as the hot favourite to win outright, she has played down the pressure.
The German, seeded 23rd, will play Polish fourth seed Agnieszka Radwanska at 3pm. They follow the 1pm meeting between French 15th seed Marion Bartoli and Belgian 20th seed Kirsten Flipkens.
Asked whether her stunning win over five-time champion Williams in the fourth round meant she was now the favourite, Lisicki responded: “Not at all, no pressure. For me, it’s still a game that I love so much, and I want to keep it that way.”
Radwanska, meanwhile, vowed to fight through the pain barrier after she suffered a thigh injury during her hard-fought quarter-final win over Li Na.
“If it’s the end of a grand slam you don’t really think about the pain, you just fight until the end,” she said.