Hewett, the number two seed and a four-time winner in New York, dropped only two games in a 6-0 6-2 victory against Spain's Daniel Caverzaschi and will face American Casey Ratzlaff for a place in the semi-finals.
Reid's match against Frenchman Stephane Houdet was a much closer contest, with the Scot eventually winning 6-3 5-7 6-2 in two hours and 13 minutes.
Up next for Reid is the defending champion and top seed Tokito Oda of Japan, who has won the past six Grand Slam singles titles.
Hewett and Reid are also attempting to win a sixth US Open wheelchair doubles title together as a pairing and on Tuesday, with a semi-final against Oda and Argentina's Gustavo Fernandez to come.
Meanwhile, Andy Lapthorne and Greg Slade were beaten in the first round of the quad singles on Wednesday.
Lapthorne lost 6-2 6-1 to top seed Niels Vink of the Netherlands, while Slade was defeated 6-4 6-2 by Gonzalo Lazarte of Argentina.
Lapthorne is still involved in the quad doubles competition and is into the semi-finals alongside South African partner Leon Els.
