Anigo's side were reeled in twice by their hosts at the Stade de la Mosson but eventually secured three points in the 89th minute when Dimitri Payet's shot was deflected over Montpellier goalkeeper Geoffrey Jourdren and into the net.
Marseille have been jeered by their fans over the past month due to poor results but Friday's win was their second in a row in Ligue 1 and kept Anigo's team in the hunt for European football next season.
"Lately, it has been quite bumpy, I am happy to have won, not against my friend (Montpellier coach) Rolland Courbis but for the players," Anigo said.
"We always have this desire to go and win everywhere...tonight we will enjoy this...tonight, the goal was to find fun."
Marseille's coach added that his team had 'deserved' to win after Mathieu Valbuena and Andre-Pierre Gignac had given them the lead earlier, only for Remy Cabella and Morgan Sanson to equalise each time for the home side.
"Every time Montpellier returned we had the desire to get the win," the 52-year-old said.
"We controlled the game and I think that victory is deserved."
Marseille stayed sixth after the win but moved level on points and goal difference with fifth-placed Lyon, while Anigo's men reduced their gap to fourth-placed Saint-Etienne to just four points.
Saint-Etienne occupy France's fourth qualifying position for next season's continental football.
Last week, Marseille supporters broke into the club's training centre and defaced buildings with demands that Anigo and president Vincent Labrune quit.
Marseille midfielder Andre Ayew urged everyone involved with the club not to get carried away with their win at Montpellier and stay focused for the last five games of the Ligue 1 campaign.
"This victory is good. Now, (we must) keep this mindset, give the best of ourselves and keep up to finish the season," the Ghanaian told Marseille's website.
"What is important for us is to take game by game and not ignite. All matches are important. We will try to take maximum points until the end."
Montpellier defender Garry Bocaly, who was a youth player at Marseille and made his senior debut there, conceded his team were disappointed at losing to Payet's late goal.
"There is a lot of frustration and anger. It's a shame to lose like that," Bocaly said.
"We know that (Marseille) need points to hang onto a European place. (Marseille) have given everything...congratulations to them."
