The Belgian edged out his Spanish counterpart 6-4 7-6(4) in one hour and 35 minutes to potentially book a date with world number one Novak Djokovic next.
World number 14 Goffin broke Verdasco - ranked 43rd - in the fifth game to establish the daylight needed to earn a first-set lead as neither service game was broken again.
Little could separate the pair for the first eight games of the second set as they sat deadlocked on four games apiece.
Goffin seemed to tilt the match in his favour by breaking to move 5-4 ahead, but Verdasco broke straight back as a tie-break loomed.
There, Goffin only lost his serve once as he eased to a 7-4 win to wrap up the match and close in on a third-round clash with Djokovic, who faces Benoit Paire tonight.