Nigeria defender Taiwo struck the winner with a low shot from just inside the area following a poor clearance by the Montpellier defence after a free-kick 10 minutes from time.
Last season Marseille went on to win the Ligue 1 title after their League Cup triumph and they will be hoping their latest success spurs them on to another championship.
"It was complicated but we expected a tough challenge," OM manager Didier Deschamps told a news conference.
"Everything is fine. We will enjoy it tonight and tomorrow and then we will focus on the league because we have a game on Wednesday (against Nice)."
Montpellier coach Rene Girard added: "Once again, we concede a goal on a set piece. We knew set pieces would be the key tonight. But I am proud of my players, they gave everything."
Marseille had the best chances but they were made to wait for their breakthrough by stubborn opponents.
The result means Montpellier have still not won a trophy since they lifted the French Cup back in 1990 when Eric Cantona and Laurent Blanc were wearing the blue and orange shirt.
MBIA INJURY
The only worry for Marseille was an injury to their Cameroon midfielder Stephane Mbia, who had to be replaced in the seventh minute, and they will wait anxiously for his scan results.
Marseille had the first clear chance in the 20th when a powerful 20-metre Lucho Gonzalez free-kick was parried by Laurent Pionnier as the holders got a grip on an ill-tempered game.
Montpellier hit back two minutes later when Jamel Saihi's attempt from five yards was saved by Steve Mandanda.
Marseille stepped up the pressure and were unlucky not to score when Mathieu Valbuena's volley hit the post in the 32nd.
They came close again just after the break when Andre-Pierre Gignac floated over a perfect cross for Andre Ayew whose header was tipped away by Pionnier.
In the 58th minute, the Ghana striker had another chance but this time his header shaved the post.
With 10 minutes left, Benoit Cheyrou unleashed a curling free-kick that the Montpellier defence failed to clear and Taiwo surged into the box to grab the winner and a deserved victory.
Last year Marseille went on to win eight of their last 10 league games following their League Cup success and clinched their first Ligue 1 title since 1992.
After the trophy presentation, Taiwo took a microphone and joined the Marseille fans in singing an anti-Paris Saint-Germain song, which includes an expletive, aimed at their arch-rivals.
Asked about the incident Deschamps said: "I did not hear it. It will give you something to write about."
