Argentina striker Dario Benedetto, with his just his fifth goal in 29 games this season, and Ahmadou Dieng found the net for Marseille.
"Qualification is welcome in the period that we are going through, we are in a tough place," said stand-in coach Nasser Larguet.
"There was not a great quality of play, but on the other hand, there was great team spirit, lots of energy, we saw the boys put in maximum effort.
"And then the satisfaction of seeing Dario Benedetto score his goal, it had been a while. The players remained united until the end."
Marseille had blown a two-goal lead to draw with Lens and lost 2-0 to arch-rivals Paris Saint-Germain since Villas-Boas was suspended following comments made after a defeat by Monaco.
The principality side also booked their spot in the last 32, as Stevan Jovetic scored in a 1-0 win at Grenoble.
Ligue 1 leaders Lille reached the next round as 19-year-old Guinean midfielder Aguibou Camara's early goal secured a 1-0 victory over Dijon.
Lille, who have not even reached the Cup semi-finals since winning the double in 2010-11, sit two points clear of Lyon at the summit of the top-flight table.
News of the sacking of Nantes coach Raymond Domenech emerged during the Canaries' 4-2 loss to Lens, with the former France boss missing the game after a positive Covid-19 test.