The nation's number one Watson dropped to a 6-2 6-4 defeat to Irina Begu while Robson, playing later in the day, suffered a spectacular collapse to lose 2-6 7-6(2) 6-1 to Sofia Arvidsson.
Watson, who recently celebrated entering the world's top 40 at 39, had chances to stamp her authority on her match against Begu but failed to convert any of the six break points she had.
Begu, ranked 53 in the world, broke three times from her six opportunities over the course of the match, which lasted an hour and 43 minutes.
The Romanian now meets Angelique Kerber in the second round after the German seed was handed a first-round bye.
"Today wasn't my best day at the office," Watson said.
Robson looked well-placed for victory over Arvidsson after a commanding opening set, during which she won 71 per cent of her first service points and saved all three break points she faced.
The 19-year-old continued in the same fashion to race to a 4-1 lead in the second set before her form completely deserted her. The experienced Arvidsson, ranked 49 in the world, took advantage to force a tie-break, which she duly claimed to send the match to a decider.
Robson, six places higher than the Swede in the world rankings, never recovered and she fell apart in the third, failing to convert any of the five break point chances she had while handing nine opportunities to her opponent.
Arvidsson took two of those, which was enough to complete victory in just over two hours on court. She goes on to meet Julia Goerges.
BNP Paribas Open results
Madison Keys (U.S.) beat Melanie Oudin (U.S.) 6-4 4-6 6-3
Garbine Muguruza Blanco (Spain) beat Bojana Jovanovski (Serbia) 2-6 7-5 6-3
Stefanie Voegele (Switzerland) beat Petra Martic (Croatia) 4-6 6-4 6-3
Magdalena Rybarikova (Slovakia) beat Romina Oprandi (Switzerland) 7-6(4) 6-0
Alexandra Dulgheru (Romania) beat Michelle Larcher de Brito (Portugal) 6-4 6-4
Daniela Hantuchova (Slovakia) beat Stephanie Foretz Gacon (France) 6-3 3-6 6-4
Mirjana Lucic-Baroni (Croatia) beat Pauline Parmentier (France) 7-5 6-3
Jamie Hampton (U.S.) beat Bethanie Mattek-Sands (U.S.) 5-7 6-3 6-1
Kiki Bertens (Netherlands) beat Annika Beck (Germany) 3-6 6-4 6-1
Alize Cornet (France) beat Monica Puig (Puerto Rico) 6-7(3) 6-2 6-2
Urszula Radwanska (Poland) beat Arantxa Rus (Netherlands) 6-3 6-3
Sofia Arvidsson (Sweden) beat Laura Robson (Britain) 2-6 7-6(2) 6-1
Kimiko Date-Krumm (Japan) beat Galina Voskoboeva (Kazakhstan) 6-4 3-6 6-4
Taylor Townsend (U.S.) beat Lucie Hradecka (Czech Republic) 3-6 7-6(1) 6-3
Irina Begu (Romania) beat Heather Watson (Britain) 6-2 6-4
Monica Niculescu (Romania) beat Elina Svitolina (Ukraine) 6-0 6-0