Britain's Joe Salisbury and his men's doubles partner Rajeev Ram continued their US Open title defence with a straight-set win to reach the third round in New York.
They had led 4-0 in the second set before Saville and Karatsev pulled back three successive games.
But Salisbury and Ram then won back-to-back games to close out the win.
Fellow Briton Lloyd Glasspool and his Finnish partner Harri Heliovaara, seeded 11th, are also through to the third round at Flushing Meadows, having beaten South Africa's Raven Klaasen and Brazil's Marcelo Melo 6-4 7-6 (8-6).
Salisbury became the first Briton to win two doubles titles at a Grand Slam in the Open era at last year's US Open.