The Briton landed only 44 per cent of his first services in and committed eight double faults but nevertheless emerged with a 6-2 6-2 win to move above Heather Watson.
The 19-year-old, who lost in the doubles final with partner Lisa Raymond in Miami on Sunday, will next face qualifier Eugenie Bouchard of Canada.
Robson, seeded 16th, need only 74 minutes for the victory after saving seven Cabeza Candela break points from seven in the first set.
In the second set she was broken by 26-year-old Cabeza Candela, but converted three of four break chances herself.
Top seed Serena Williams moved into the third round with a 6-2 6-3 win over Camila Giorgi of Italy and will next face Sabine Lisicki, who double-bagelled Anna Tatishvili.
The top seed fired nine aces and broke her opponent five times.
"It definitely wasn't great for me - I played well, but I wasn't great, footing-wise," Williams said. "I was just trying to get my stability. I only had one or two days of practice. But I felt good - I'm glad I came here and hope I can get ready for what can hopefully be another good clay court season.
"I've played her in doubles before and I think she's a good player. I also saw she did really well at Wimbledon - I knew one day I would play against her, so I kind of knew how she would play going into the match."
Sloane Stephens, seeded fourth, continued her post-Australian Open slump as she fell to a 6-2 6-0 defeat to fellow American Bethanie Mattek-Sands. Stephens has won just twice since reaching the semi-finals in Melbourne in January.
Seeds Carla Suárez Navarro and Mona Barthel was also ousted in the second round.
At the WTA tournament in Monterrey, second seed Marion Bartoli was upset by Coco Vandeweghe of the US, the French world number 14 falling to a 4-6 6-3 6-1 first-round defeat.
Bartoli committed 19 double faults in an abysmal serving display as Vandeweghe scored what is probably the biggest win of her career so far.
There were no problems for the other top seeds Angelique Kerber, Ana Ivanovic and Maria Kirilenko, who won in straight sets to reach the second round.
Family Circle Cup results
1-Serena Williams (U.S.) beat Camila Giorgi (Italy) 6-2 6-3
Bethanie Mattek-Sands (U.S.) beat 4-Sloane Stephens (U.S.) 6-2 6-0
Stefanie Voegele (Switzerland) beat 7-Carla Suarez Navarro (Spain) 6-2 6-4
Jessica Pegula (U.S.) beat 8-Mona Barthel (Germany) 7-6(4) 6-1
11-Sorana Cirstea (Romania) beat Anabel Medina Garrigues (Spain) WO
12-Varvara Lepchenko (U.S.) beat Mathilde Johansson (France) 6-1 4-6 6-1
9-Jelena Jankovic (Serbia) beat Melanie Oudin (US) 6-4 5-7 6-4
10-Julia Goerges (Germany) beat Yulia Putintseva (Kazakhstan) 6-2 6-3
Vania King (US) beat 14-Yaroslava Shvedova (Kazakhstan) 4-6 6-1 6-4
15-Sabine Lisicki (Germany) beat Anna Tatishvili (Georgia) 6-0 6-0
16-Laura Robson (Britain) beat Estrella Cabeza Candela (Spain) 6-2 6-2
Mirjana Lucic-Baroni (Croatia) beat Romina Oprandi (Switzerland) 4-6 6-1 6-3
Madison Keys (US) beat Alexandra Cadantu (Romania) 7-6(3) 6-4
Marina Erakovic (New Zealand) beat Daniela Hantuchova (Slovakia) 7-6(2) 6-4
Monterrey Open results
1-Angelique Kerber (Germany) beat Samantha Crawford (U.S.) 6-3 6-3
Coco Vandeweghe (U.S.) beat 2-Marion Bartoli (France) 4-6 6-3 6-1
3-Maria Kirilenko (Russia) beat Ximena Hermoso (Mexico) 7-5 6-1
4-Ana Ivanovic (Serbia) beat Marta Sirotkina (Russia) 6-0 6-0
5-Anastasia Pavlyuchenkova (Russia) beat Nina Bratchikova (Russia) 6-4 6-0
8-Ayumi Morita (Japan) beat Jana Cepelova (Slovakia) 6-0 6-1
Jovana Jaksic (Serbia) beat Vera Dushevina (Russia) 4-6 7-6(4) 7-6(2)
Alla Kudryavtseva (Russia) beat Eva Birnerova (Czech Republic) 6-4 6-3
Tereza Mrdeza (Croatia) beat Daria Gavrilova (Russia) 6-4 6-2
Donna Vekic (Croatia) beat Julia Cohen (U.S.) 6-0 6-4
Lauren Davis (U.S.) beat Olga Puchkova (Russia) 6-2 (Puchkova retired)
- Sports & Recreation