Robson teamed with Australia's Sally Peers to beat experienced American duo Jill Craybas and Abigail Spears 6-3, 7-5.
It was a good morning for the British women in doubles action with Scotland's Elena Baltacha and Sarah Borwell also progressing to the second round in their respective matches.
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Hide AdBaltacha followed her upset victory over 30th seed Kateryna Bondarenko in the singles yesterday by partnering Liga Dekmeijere to a 7-5, 6-2 victory over Lucie Safarova and Aleksandra Wozniak.
Borwell then completed a hat-trick of win alongside American Raquel Kops-Jones.
The pair defeated Polona Hercog and Petra Martic 6-4, 7-6 (7/3).
In the women's singles, sixth seed Venus Williams romped through to the third round with a comfortable 6-2, 7-5 victory over Austria's Sybille Bammer.
Denmark's Caroline Wozniacki, the fourth seed, enjoyed a 6-3, 6-1 win over Germany's Julia Goerges.
Former finalist Ana Ivanovic, of Serbia, (No.20) was beaten 6-7, 7-5, 6-4 by the unseeded Gisela Dulko.