Fabregas, who joined the Principality club from Chelsea on Friday, struck a composed figured in the midfield as Thierry Henry's side recovered from a shaky start to earn a point thanks to Youri Tielemans' leveller six minutes before the break.
The World Cup winner almost laid on a winner for Tielemans in added time with a trademark clipped pass but the Belgium international hesitated on the ball and ended up shooting a presentable opportunity to snatch a rare victory well wide.
Fabregas' arrival on the Mediterranean coast on Friday was the third in January after veteran defender Naldo and France under-21 left-back Fode Ballo-Toure, who both acquitted themselves well on Sunday.
The draw was a good result for a team without a fit striker and not enough available players to even fill the substitutes' bench.
They remain in the relegation zone despite the positive performance and are four points from safety after winning just twice in 10 league games since Henry's arrival in October.
However they have been slightly helped in the quest for survival by Paris Saint-Germain battering fourth-from-bottom Amiens 3-0 on Saturday and Caen in 16th losing 3-1 at home to Lille.
- Marseille protests -
Hosts Marseille began the match amid protests from fans about their own poor form which sees them down in ninth, 22 points behind runaway leaders PSG.
Rudi Garcia's side are winless in eight in all competitions and as well as being well back in the league are out of all three cups after being humiliated 2-0 by fourth-tier Andrezieux in the French Cup last week.
They were eliminated from the League Cup last month by Strasbourg and finished an embarrassing Europa League group stage with just one point.
Fans head up a banner that read "owners, coach, players... all culpable" and barely celebrated Maxime Lopez's 13th minute opener, which squeezed under a poor effort to save the shot from Diego Benaglio.
Midway through the second half they shouted "ole" when Monaco kept possession and whistled their own players when they won the ball back.
When Thuavin netted what he thought was the winner 20 minutes from the end he remonstrated with supporters behind the goal who refused to cheer, drawing loud boos that were only beaten in volume when referee Mikael Lesage ruled the goal out for a foul from Lucas Ocampos on Benaglio.
Earlier on Sunday Montpellier slipped further back from the Champions League places with a 1-1 draw at third-from-bottom Dijon, while Strasbourg continued their claim for a European place with a 2-1 win at Toulouse.