Andy Murray made the most of his wildcard entry to the Cincinnati Masters with an opening-round 6-4, 6-4 triumph against Richard Gasquet.
20-time Grand Slam champion Federer beats Gasquet in straight sets. Elsewhere, Daniil Medvedev has it easy against Carlos Alcaraz.
The 20-time Grand Slam champion is through to the second round after Mannarino called it quits at the start of the fifth set.
Popyrin did hold the advantage in the third set, as he led 5-2, but Nadal pulled it back to 5-5 before ending the match in a tie-breaker.
The three highest-seeded players were all eliminated from the Emilia-Romagna Open on Wednesday (May 26).
The Greek player won the match in straight sets to reach the quarterfinals, where he will play Yoshihito Nishioka.