Wild-card Watson had too much for Hungary's world number 106 Greta Arn, converting her only two break point opportunities of the match while avoiding any on her own serve as she came through 6-3 6-4.
And Robson showed plenty of resolve as she fell behind to world number 49 Maria Jose Martinez Sanchez of Spain before battling back to win 3-6 6-2 6-2, breaking her opponent four times in the last two sets.
On a day of upsets, top seed Agnieszka Radwanska and second seed Petra Kvitova were both knocked out, while former world number one Caroline Wozniacki also suffered an early exit.
Radwanska was beaten 6-2 6-4 by Bulgaria's former Wimbledon semi-finalist Tsvetana Pironkova, while reigning Wimbledon champion Kvitova went down 7-5 6-4 to Ekaterina Makarova of Russia.
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Third seed Wozniacki's stated hope of contending for the Wimbledon title looks more unlikely than ever after she came out on the wrong end of a 6-1 6-7 (9/7) 6-4, two-and-a-half hour defeat to American Christina McHale.
Fourth seed Marion Bartoli and fifth seed Angelique Kerber avoided upsets with wins over Sorana Cirstea and Elena Vesnina respectively while Stephanie Dubois, Lucie Safarova, Klara Zakopalova and Anastasia Pavlyuchenkova were also winners.
The upsets continued in the second round of the men's event, with top seed Richard Gasquet dumped out by Australian world number 82 Marinko Matosevic, who rebounded from a nightmare first set to go through 1-6 7-6 (7/5) 7-6 (7/3).
In first round action, Andy Roddick beat American compatriot Sam Querrey when the latter retired injured trailing 5-2 in the first set. Other winners were Steve Darcis, Philipp Kohlschreiber, Denis Istomin and Yen-Hsun Lu.