Andre Ayew headed in Mathieu Valbuena's corner in the 93rd minute to send the hosts into delirium with the late winner.
Though the visitors had opportunities, namely through Diego Forlan's first-half effort which was tipped over by Steve Mandanda. Subsequent chances from Mauro Zarate and Esteban Cambiasso failed to materalise, as the Nerazzuri ultimately fell to a sixth defeat in seven games to put Claudio Ranieri's tenure as Inter boss in serious doubt.
"It's good to have this success at the very end. We were against a great team tonight but I think we had more control," the 43-year-old was quoted as saying by uefa.com after the match.
"It was a game of few chances but we were the better side at the end. We have a small advantage. It's better than last year [when Marseille drew 0-0 with Man Utd at this stage] though the second leg will be tough too."
Goalscorer Andre Ayew also gave his thoughts to reporters after his late goal. "It was a good result at home. This victory makes us feel good. We must continue to work, work, work," the Ghana international told TF1.
"Our fans are happy and we will do everything to make them even happier in all competitions."
Following the victory, Marseille encounter Inter in the second leg at San Siro on March 13. Both sides return to league action at the weekend, with L'OM visiting Stade Brestois as Claudio Ranieri's men tackle Napoli in Serie A.
How do you stay up with football when on the move? With http://m.goal.com – your best source for mobile coverage of the beautiful game.