Costa, seeded 26, had a comfortable victory over Belgium's Christophe Rochus 6-1, 6-2, 7-5 and he will face another Belgian, Xavier Malisse.
In the women's draw, 2001 champion Capriati made it through as did 2000 winner Mary Pierce of France and fourth seed Venus Williams, who will face each other in the third round.
Serena Williams, the 2002 champion, had to come back from a set down to beat 17-year-old Russian Maria Kirilenko winning 6-4, 2-6, 6-4 and she next faces Croatia's Silvija Talaja.
Capriati survived an early wobble to defeat qualifier Kveta Peschke of the Czech Republic 7-5, 6-3 and now faces Russian 25th seed Elena Bovina.
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