The 17 year-old, seeded 14, defeated Oudin 6-3, 6-4 to set up a meeting with Italy's Anna Floris.
The Melbourne-born Briton, who reached the second round of Wimbledon and the US Open last year, broke four times, dropping her serve twice.
Naomi Broady joined Robson in the second round as she saw off the challenge of another American, Ashley Weinhold, 6-4, 5-7, 6-3.
Ward, who is the 32nd seed, lost the first set against Iñigo Cervantes-Huegun of Spain on Wednesday before the match resumed with the Briton in much better touch.
He won successive sets on court four to seal a 6-7 (2/7), 7-6 (7/3), 6-1 win and the British No 2 will now meet Czech Ivo Minar in the next round.
Irishman Conor Niland went out after losing his first-round match to Switzerland's Stephane Bohli 6-3, 6-2.