Fabbiani, nicknamed "The Ogre" after the animated film character Shrek, came on as a substitute in the 57th minute with River 1-0 down and the game immediately turned.
Martin Galmarini headed River level five minutes later before Fabbiani, signed from Central's arch-rivals Newell's Old Boys earlier this month, rifled in the winner with a 25-metre shot in the 67th minute.
Milton Carraglio had given Central the lead after 15 minutes.
River, who finished bottom of the Apertura tournament in the first half of the season, badly needed the win after squandering a 2-0 lead against modest Colon to draw their opening game 2-2 last week.
In other matches, Cristian Cano scored a second-half hat-trick to give San Martin-Tucuman a 3-0 win over Independiente while Godoy Cruz overcame some slapstick defending to beat San Lorenzo 2-1.
San Lorenzo went ahead in the third minute when Godoy Cruz defender Gabriel Valles blasted a botched clearance straight into the face of his goalkeeper Jorge Carranza and the ball rebounded into the net.
However, the game turned when San Lorenzo defender Gaston Aguirre was sent off for a second bookable offence in the 38th minute.
Godoy Cruz made the advantage pay as second-half goals from Ivan Borghello and Leandro Caruso earned them maximum points.
Newell's Old Boys, Huracan and Arsenal are the only teams with six points from their opening two games.
Huracan won 4-1 at Racing Club on Friday, while Newell's beat Boca 2-0 on Saturday and Arsenal beat Tigre by the same score.
