0302 GMT October 14, 2019
Fekir scored twice and won a penalty that Alexandre Lacazette, who also grabbed a double, converted to put Lyon on 57 points from 28 games, one ahead of second-placed Paris St Germain which thumped RC Lens 4-1 on Saturday.
The other goal for the league leaders came from Corentin Tolisso in added time.
"It shows that when we are at our best physically and mentally we can do a lot of damage," Lacazette told Canal Plus.
Montpellier, seventh on 42 points, went in front early on through Lucas Barrios who also missed a penalty nine minutes from time.
Barrios tapped the ball in from close range after being set up by Bryan Dabo in the sixth minute but Lyon did not panic.
Fekir, at the heart of a mix-up over his choice of national team between Algeria and France, was brought down in the box on the half hour.
Ligue 1 top scorer Lacazette knocked in the resulting penalty for Lyon which lost 2-1 at Lille last week.
After a fine one-two with Yoann Gourcuff, Fekir fired home a bullet shot from just outside the box five minutes before the interval.
Gourcuff then set up Fekir again and the midfielder made it 3-1 with a delicate chip in the 72nd minute before Barrios' penalty was blocked by Anthony Lopes.
Tolisso rubbed salt into Montpellier's wounds and there was still time for Lacazette to add his 23rd league goal of the season deep into added time.
"We did not start well but quickly felt this game could be ours," said midfielder Maxime Gonalons.