The Frenchman took an hour and 23 minutes to beat the 38th-ranked player 6-3 6-2 in Toronto.
Gasquet had to save four break points in the first set, but needed just one of the two that he created on the Slovak's serve to take a one-set lead.
Once again, the seventh seed proved unbreakable in the second set as he prevented Klizan three more times. However, Gasquet was far more creative in his play, setting up 10 chances.
He managed to take two of them on his way to wrapping the set up 6-2 and booking a clash with either Kei Nishikori or Andreas Seppi.