After just over an hour and a half, fifth seed Evans sealed a 6-4 6-4 victory over Egor Gerasimov in their first meeting on the ATP Tour.
Evans won 85 per cent of points behind his first serve and did not face a single break point as he advanced to the second round. Malek Jaziri or Alejandro Davidovich Fokina will be his next opponent.
Following a tight opening set, Khachanov overcame Mackenzie McDonald 7-6 (7-3) 6-3 to continue his impressive form early in the 2022 season.
Alex Molcan came from a set down and navigated a second-set tie-break to defeat Chris O'Connell and earn a meeting with Shapovalov in the second round.
At the Open 13 in Marseille, home hope Richard Gasquet overcame Mikael Ymer 6-4 7-5 to book a meeting with Andrey Rublev in the second round. There were also wins for Stefano Travaglia and Ilya Ivashka.