Murray next 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.
With Verdasco serving first and no breaks, Murray was serving to stay in the first set at 5-4 down.
But on set point for his opponent, the Scot served his first double fault -- missing badly -- to hand the Spaniard the first set.
With the home 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 for 3-3 as Murray hit a string of unforced errors.
Verdasco broke again for 5-3 with a lucky backhand return that hit the net and dropped over.
In the next game, Murray hit a dazzling forehand winner to finish a 28-stroke rally that had him scampering from side to side, on his way to giving himself 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 he 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 fought off two break points in the sixth game to level at 3-3. The Briton broke in the next game then served to love 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.
The Spaniard challenged and the two players smiled at the net as they waited for the replay result but there was little doubt that the call was going Murray's way, sending waves of relief and elation around Centre Court.