Second seed Murray won 6-4 6-1 6-4 under the new roof on Arthur Ashe Stadium as a storm hit New York.
Evans beat German Alexander Zverev 6-4 6-4 5-7 6-2 to reach the third round for the first time in three years.
Murray now faces Italian Paolo Lorenzi while Evans faces Swiss Stan Wawrinka.
Edmund, 21, reached the third round with a straight-sets win on Wednesday.
Paul Hutchins, Roger Taylor and Peter Curtis were the last three British players to make the third round of the US Open in the same year, in 1968.
Sixth seed Kei Nishikori of Japan beat Russia's Karen Khachanov 6-4 4-6 6-4 6-3 and Austrian eighth seed Dominic Thiem saw off Ricardas Berankis of Lithuania 6-4 6-3 6-2.