Javier Pastore scored the only goal five minutes into the second half to put leaders PSG on 73 points, nine ahead of second-placed Olympique Marseille with four matches left.
Beckham, who came on as a second-half substitute, was shown a straight red for a foul on Youssef Adnane two minutes into stoppage time after Marco Verratti had been sent off in the 80th when he picked up a second yellow card.
A mass brawl erupted after the final whistle and Evian's Saber Khlifa was shown a red card by referee Olivier Thual, who also asked for PSG goalkeeper Salvatore Sirigu to join him back on the pitch.
"This kind of thing happens in a football game, tempers flared a little bit, that's all," PSG defender Christophe Jallet told French television Canal Plus.
Olympique Lyon stayed third on 60 points after Yoann Gourcuff's second-half strike salvaged a 1-1 home draw against arch rivals Saint-Etienne earlier on Sunday.
Zlatan Ibrahimovic's fierce 25-metre shot was saved by Bertrand Laquait as PSG got off to a good start.
Evian had their first chance in the eighth minute when Khlifa's dipping shot from the edge of the box went just over Sirigu's crossbar.
Laquait was then at it again to deny Ibrahimovic's soft volley from Maxwell's free-kick.
PSG's Ezequiel Lavezzi later showed another example of his clumsiness by fluffing a shot after collecting a brilliant pass from Ibrahimovic.
The Sweden striker, in front of an open goal after Verratti's shot had bounced off the bar and into his path, skied his attempt over the bar while his Argentine team mate inexplicably failed to find the net after dribbling past Laquait.
Sirigu then pulled off a brilliant save to deny Khlifa's powerful shot and PSG remained under pressure, especially after Verratti was sent off.
