American Rogers, the 21-year-old American, thrashed Czech qualifier Tereza Martincova 6-1 6-3 in Quebec City.
Rogers was largely untroubled in the 65-minute encounter at the WTA International event played on indoor carpet.
She broke serve five times to set up a second-round clash with qualifier Asia Muhammad.
American Muhammad needed two hours 33 minutes to edge Brit Johanna Konta 6-2 6-7 (0-7) 7-6 (7-4).
Goerges, the German fifth seed, sent down 10 aces in a 6-1 6-2 victory over local hope and wildcard Stephanie Dubois.
Dubois was only able to win 12 return points against the hard-hitting Goerges.
The only seed to fall was American seventh seed Anna Tatishvili, who was beaten 7-6 (11-9) 6-4 by German Tatjana Maria.
In an all-American affair, Madison Brengle proved too good for qualifier Samantha Crawford with a 6-3 6-3 victory.
Kazakh Sesil Karatantcheva also advanced, battling past American qualifier Sanaz Marand 3-6 6-1 7-6 (8-6) after coughing up a 4-0 third-set lead.