Both players managed to secure breaks of serve in the early stages of the match, but Haase was able to capitalise and earn a second to take the first set 6-4.
The number 12 seed hit back in the second set as he improved on his serve to fend off the threat of the Dutchman, and then secured a break that allowed him to tie the game at one set apiece courtesy of a 6-3 win.
The duo were involved in a tense battle in the third set as they exchanged breaks of serve once again, which sent the set to a tie-breaker. However, Gasquet cruised to victory as he lifted the quality of his game to take the lead in the match.
Gasquet looked to have secured a vital break in the fourth set to give him the chance to close out the match, although Haase responded to hit back, but his attempts to level the scores were thwarted when the Frenchman broke again to clinch his place in the third round.
Gasquet will face Bernard Tomic in the last 32 of the competition.