British number two and world number 50 Watson, 23, lost 6-2 6-3 to Russian Elena Vesnina, ranked 53.
She was broken in the sixth game of the first set and then the eighth game of the second in gusty conditions.
British number four Moore, also 23, failed to convert five set points in the first set before losing 7-5 6-4 to Ekaterina Makarova of Russia.
The world number 229 led 5-3 in the opener before Makarova fought back to take the set and then the second after breaking in the fifth game.
British number three Naomi Broady, 26, was defeated 6-1 6-1 by Puerto Rico's Monica Puig.
Johanna Konta, Britain's number one, is one of the 16 seeds and the 25-year-old has a bye into the second round.
Polish former champion Agnieszka Radwanska, who is the top seed, and two-time Wimbledon winner Petra Kvitova of the Czech Republic will also enter the competition at the second-round stage.
