The Spanish midfielder, who joined from Chelsea last month, fired home in the second half to help Monaco move from 19th to 18th in the standings and claim their first victory at home in the league.
They have 18 points from 23 games and are level on points with 17th-placed Caen who have a game in hand.
In coach Leonardo Jardim's second game since he returned to the bench after Thierry Henry was sacked last week, Monaco showed some steel.
"I have congratulated the players for the first three points of this mini-league. Now we have to prepare for the next game," said Jardim.
"The last minutes were difficult. Even if we manage to forget the past, we hadn't won in a long time and it can get a little scary."
Monaco opened the scoring after 15 minutes when Aleksandr Golovin latched on to a Gelson Martins cross, only for Christopher Jullien to level five minutes later from close range.
Fabregas, however, restored the advantage with a right-footed striker from the spot that went under the bar in the 62nd minute.
In Jardim's first game in charge earlier this week, Monaco were knocked out of the League Cup semi-finals by Guingamp on penalties after a 2-2 draw.
Olympique de Marseille dropped down to 10th on 31 points after they lost 2-1 at Stade de Reims, with Clinton Njie's last-gasp goal being too little, too late after Tristan Dingome and Suk Hyun-jun had put the hosts in a comfortable position.
(Reporting by Julien Pretot; Editing by Clare Fallon)
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