The former England captain was shown a straight red card in the last minute - six minutes after being introduced - for a dangerous lunge on Youssef Adnane, meaning PSG ended the game with nine men following the earlier dismissal of Marco Verratti.
But Leonardo claimed after the game that Beckham had been the victim of an error by match official Olivier Thual.
"The referee lacked lucidity," the Brazilian told Le Parisien.
"He made mistakes. The red card shown to Beckham, for example, was really unfair.
"It was always a very tense match. Nobody wanted to lose - especially not Evian, who are struggling to stay up.
"I also believe that our team is really tired at the end of the season."
Evian striker Saber Khelifa and PSG goalkeeper Salvatore Sirigu were involved in an altercation which saw both also red-carded as skirmishes ensued after the full-time whistle.
"I did not see much of the fight," Leonardo added.
Coach Carlo Ancelotti said: "There was a little scuffle, I do not know, I was not there."
During the match, Javier Pastore's 50th-minute goal proved decisive as PSG re-established their nine-point advantage over second-placed Marseille with four games left.