Beckham was sent off just six minutes after going on as a substitute, with his side already a man down after Marco Verrati's red card.
Beckham was shown a straight red card deep into stoppage time following a dangerous lunge on home substitute Youssef Adnane.
In the aftermath, PSG assistant coach Claude Makelele was sent to the stands for remonstrating and emotions continued to run high after the final whistle, when Evian striker Saber Khelifa and PSG goalkeeper Salvatore Sirigu were involved in a scuffle which saw both also dismissed.
The scenes overshadowed the rest of the match, during which Javier Pastore's 50th-minute goal had proved decisive as PSG re-established their nine-point advantage over second-placed Marseille with four games left.
Behind PSG and second-placed Marseille, Lyon are in third and in pole position to stay there after a 1-1 draw with fourth-ranked Saint-Etienne.
A second-half Yoann Gourcuff strike cancelled out Kurt Zouma's opener for Les Verts as Lyon missed out on a third successive league win but stayed on course for Champions League qualification.
The visitors were guilty of wasting a succession of chances and will be disappointed to not get the win which would have seen them leapfrog their rivals.
Lyon are feeling the heat, though, as Nice came from behind to beat bottom club Troyes 3-1 and keep the pressure on.
Despite trailing to a 21st-minute Mohamed Yattara goal, Claude Puel's men clawed their way back thanks to goals from Renato Civelli, Dario Cvitanich and Luigi Bruins.