The 29-year-old, who can end the week as world number one if he wins the tournament and Novak Djokovic fails to make the final, battled to a 6-3 6-7(5) 7-5 victory in just over two-and-a-half hours.
It was a comfortable opening set for the Scotsman as he wrapped it up in 31 minutes with a single break of serve, but he faced a fightback from Verdasco in the second.
Murray held his opening service game by saving two break points, but he lost his next one as Verdasco took a 3-1 lead on the hard court.
The three-time Grand Slam champion led a comeback of his own as he managed to pull the set back to 5-5 before forcing it into a tie-break, which Verdasco clinched 7-5.
Both players dropped serve early in the third and deciding set, and it was a frustrating end for Murray as he squandered six break points before eventually taking serve and booking himself a place in the next round.