Verdasco raced into a two-set lead against the world number two, but Murray managed to haul himself back into the match, eventually winning 4-6 3-6 6-1 6-4 7-5 to reach his fifth consecutive semi-final at SW19.
Murray had nothing but praise for Verdasco's performance on the day, reserving a special mention for his serve as well as acknowledging the atmosphere created by a vociferous crowd.
"He served unbelievably well especially when he was behind. In the first set he played some really good stuff, in the second set my level dropped and I started rushing a bit but just managed to turn it round," Murray told BBC Sport.
"I made some poor choices. He served fantastically well. He's a very, very good player he's been at the top of the game before and he's playing well again. I started to play more solid and really took my time when I had the chance.
"There's been a lot of matches where I've been behind and managed to turn it round; I don't know if it is the most emotional match, but it was an unbelievable atmosphere and great to get through."
Murray will now face Jerzy Janowicz in Friday's semi-final.