The third-seed Wawrinka, who reached the semi-finals last year, defeated Spaniard Fernando Verdasco 7-6 (4), 6-4, 6-4 in 2 hours and 17 minutes.
The Swiss player survived four break points and broke Verdasco twice.
On the other hand, Ferrer eased past Ukraine's Alexandr Dolgopolov, who was trailing 6-5 in the first set when he retired due to an injury.
Another Spaniard, 16th-seeded Feliciano Lopez won in similar fashion, falling behind 3-4 in the first set to Borna Coric before the Croatian was forced to retire because of injury.
In the women's category, fifth-seed Simona Halep and 11th seed Carla Suarez Navarro advanced to the second round.
Halep rolled over Belgium's Kirsten Flipkens 6-0, 6-2 to claim her spot in the second round while Suarez Navarro, playing in her ninth US Open, dominated Teliana Pereira of Brazil 6-0, 6-0 to advance.