The French ninth seed squandered three break points in the opening set before converting the fourth to take it 7-5.
Spain's Verdasco moved into a 5-3 lead in the second as he looked on course to level the match, but Gasquet fought back to force the tie-break.
However, Verdasco resisted another resurgence from his opponent to clinch the tie-break 8-6.
Gasquet looked in control during the decider but struggled to capitalise on any of the five break points that he brought up.
Despite those missed opportunities, he broke to go 4-3 ahead before closing out the match 6-3.