The Uruguay striker, who joined the club in 2013, struck in the 11th minute as he tapped in from Adrien Rabiot's cross to move past the record of Swede Zlatan Ibrahimovic with his 157th goal in his 229th game in all competitions for PSG.
Neymar, who had missed the three previous games through injury, found the back of the net each way of the interval and Angel Di Maria also scored, putting PSG on 59 points from 23 games.
Second-placed Olympique Lyonnais, on 48 points, travel to Girondins de Bordeaux and third-placed Olympique de Marseille take on Monaco, fourth, on Sunday.
At the Parc des Princes, Montpellier entered the pitch with the best defence in the league with 15 goals conceded in 22 matches, but their back four were torn up by the home forwards.
Cavani beat Ibrahimovic's record of 156 goals for PSG from close range before Neymar doubled the tally from the penalty spot after Pedro Mendes was sanctioned for handball in the area five minutes from the break.
Referee Frank Schneider congratulated Cavani on his record but still showed him a yellow card for taking off his shirt in celebration.
Di Maria added a third from Thomas Meunier's cross with 20 minutes left as PSG continued to pile on the pressure.
Neymar scored the last goal with his knee as he latched onto Cavani's cross eight minutes from time.
"I've come here to make history, give my best and score a lot of goals," said Neymar, who added he had been chosen by coach Unai Emery as the one who would be taking the penalties for PSG.
"The coach chose me to take the penalties and there is no problem with that."
PSG have now scored 72 league goals, more than any team in the five major European leagues.
(Reporting by Julien Pretot; Editing by Christian Radnedge)
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