Gasquet, who made the finals of the Rogers Cup in Toronto on Sunday, was unable to maintain any momentum against Raonic's attack.
"I was hoping to play a little bit better last week," said Raonic, referring to his early exit from the Rogers Cup. "Coming here I'm playing better. I still have a long ways to go, but I feel like I'm on the right track.
"I just need to do a few things better, and I think I will be setting myself on the right course."
Raonic, from Thornhill, Ont., will next face former Australian Open finalist Marcos Baghdatis, who beat Alejandro Falla of Colombia 6-4, 6-3.
Raonic saved all four of the break points he faced in the 90-minute win on Tuesday, a day after reaching his career-high No. 19 ranking which assures him a solid ranking when the US Open begins Aug. 27.
"I need to create a little bit with my serve," said Raonic. "I was able to get ahead (Tuesday), but I wasn't picking up enough free points. I wasn't hitting the spots exactly."
Raonic will try to improve his record to 2-0 over Baghdatis, whom he beat indoors in Stockholm last October in a first round.
The Canadian said he need to exploit his game more if he is to make a breakthrough into the ATP top 10 and beyond.
"There is always a window (of opportunity). I have seen openings in let's say the window, but I have sort of just been like sliding it open slowly," said Raonic. "I just need to sort of kick through and go for it and just try to make the most of it. I think there will be opportunities.
"Lately it's been pretty predictable within the top three and top four. It's tennis, it's sport, you never know."