Davis Cup star Evans beat Canada's Peter Polansky 6-2 6-3, while Murray beat Australian Arina Rodinova 6-4 6-1 to set up a second-round qualifying clash with American teen Taylor Townsend.
Townsend, 17, reached the junior final of this year's Wimbledon but has never played in a main Grand Slam event. She beat qualifying 11th seed Eva Birnerova of the Czech Republic 6-1 6-2.
Moore, however, could not get past Colombia's Catalina Castano, going down 6-4 1-6 6-4.
Another notable qualifier for the second round was American 21-year-old Coco Vandeweghe, a former US Open junior champion and four-times veteran of the main draw.
James Ward and Elena Baltacha made it through to the penultimate round of qualifying on Wednesday, meaning four Britons are two matches away from joining Andy Murray, Laura Robson and Heather Watson in the main draw.
The US Open takes place at Flushing Meadows between August 26 and September 9.
- Sports & Recreation
- Peter Polansky
- Eva Birnerova