The Westmount, Que., native won by scores of 6-4, 3-6, 6-1 over the Briton, who has been her friend for years since they both rose in the junior ranks. The second round match between the 19-year-olds at the Family Circle Cup lasted just over two hours.
Bouchard broke her opponent's serve seven times.
Bouchard is in her first full year on the pro circuit after winning Wimbledon as a junior in 2012, which made her the first Canadian ever to win a junior singles Slam.
Last month, she upset longtime ranked player Shahar Peer before falling to Maria Sharapova in Key Biscayne, Fla.
Robson, ranked 43rd, also won Wimbledon as a junior player.
Bouchard is ranked 114th, but the number is expected to change for the better in the coming weeks.
The five-foot-ten Bouchard will face 2011 U.S. Open winner Samantha Stosur, who dispatched Marina Erakovic of New Zealand by scores of 6-1, 6-2.
In other action, second-seeded Caroline Wozniacki opened with a strong showing Wednesday, defeating Silvia Soler-Espinosa 6-1, 6-1 on Wednesday.
Wozniacki only entered the tournament Saturday when a scheduled mission trip to Haiti with boyfriend Rory McIlroy was postponed. The 2011 Family Circle champion didn't show much rust from her last-minute decision to play. She broke Soler-Espinosa twice to win the final five games of the opening set.
In other early matches, 2009 Family Circle champion Sabine Lisicki fell to American qualifier Mallory Burdette in three sets and No. 10 Julia Goerges ousted Olga Govortsova in straight sets.