The 16th seed, who will overtake Heather Watson as British number one next week, was favourite against an opponent who is 72 places lower than her in the WTA world rankings (114 to Robson's 42).
The first set was a long, messy affair, but Bouchard dug in to win it 6-4.
At one stage during the second set, Robson shouted to herself "Stop being such a muppet" and then after another miss said: "Again, double muppet."
However she fought back to win that second set before fading badly in the third against the more consistent Canadian.
Such was Robson's frustration that she threw her racquet at her chair when the match finally finished.
There was no such drama for second seed Caroline Wozniacki though as she defeated Spain's Silvia Soler Espinosa 6-1 6-1 in her second round contest.
"I felt like I served really well today, and returned really well too," Wozniacki said.
"Playing your first match of the year on clay, it's always going to be difficult to slide and move unbelievably, but I felt like I did a pretty good job of that out there and I'm pleased with my work."
Australian third seed Samantha Stosur crushed Kiwi rival Marina Erakovic 6-1 6-2 to seal her progress into the next round.
"I'm really pleased with the way I went out there and played today," Stosur said.
"I went for my serve a lot and whenever I had the opportunity I tried to get the ball off the court. That's why it really worked so well and helped me think out there, 'Okay, this is going well, I've got to keep at it.'
"I probably didn't expect to win my first match back (from injury... But as the match went on I felt really good and the way I was playing all came together. And of course I really enjoy playing on this court, so I always try to do my best every time I'm out there.
"I mean it's all great but I don't want to get ahead of myself. I certainly want to take it day by day, keep doing the right things."
Venus Williams was made to work hard for her place in the third round after world number 104 Monica Puig took her to three sets.
Fifth seeded Williams appeared to be cruising to victory after romping through the opening set, but the Puerto Rican fought back to claim the second.
Williams broke twice in the decider to eventually wrap up a 6-2 5-7 6-3 victory.
"She really found her rhythm in the second set," Williams said afterwards.
"She started controlling the points, so in the third set I just tried to play confident and hopefully get some first serves in. But she played really, really well - she should be looking forward to a lot of great wins in the future."
Ninth seed Jelena Jankovic won through, but was also pushed to her limits in a 5-7 7-6 6-3 win over Caroline Garcia of France.
Family Circle Cup results
5-Venus Williams (U.S.) beat Monica Puig (Puerto Rico) 6-2 5-7 6-3
9-Jelena Jankovic (Serbia) beat Caroline Garcia (France) 5-7 7-6(10) 6-3
6-Lucie Safarova (Czech Republic) beat Mirjana Lucic-Baroni (Croatia) 7-5 6-1
Andrea Petkovic (Germany) beat Vania King (U.S.) 7-5 6-4
3-Samantha Stosur (Australia) beat Marina Erakovic (New Zealand) 6-1 6-2
Mallory Burdette (U.S.) beat 15-Sabine Lisicki (Germany) 3-6 7-6(5) 7-5
2-Caroline Wozniacki (Denmark) beat Silvia Soler Espinosa (Spain) 6-1 6-1
Eugenie Bouchard (Canada) beat 16-Laura Robson (Britain) 6-4 3-6 6-1
10-Julia Goerges (Germany) beat Olga Govortsova (Belarus) 6-2 0-6 6-3
Madison Keys (U.S.) beat Grace Min (U.S.) 6-4 6-3
Monterrey Open results
Monica Niculescu (Romania) beat 6-Yanina Wickmayer (Belgium) 6-4 3-6 6-1
7-Urszula Radwanska (Poland) beat Donna Vekic (Croatia) 6-4 6-1
8-Ayumi Morita (Japan) beat Galina Voskoboeva (Kazakhstan) 6-4 7-6(3)
5-Anastasia Pavlyuchenkova (Russia) beat Tereza Mrdeza (Croatia) 6-1 6-1