The hosts looked to be heading for a first league win of the season as they took 2-0 lead into the break but Monaco fought back and Fabinho scored from the spot with seconds remaining after goalkeeper Jonathan Ligali brought down Fabio Coentrao.
The victory took Leonardo Jardim's Monaco up to 10th on 11 points from seven games, six adrift of leaders Paris St Germain, while Montpellier are bottom with just one point.
Home defender Daniel Congre had opened the scoring after 26 minutes when he netted from close range after Hilton's header from Ryad Boudebouz's corner was saved by Danijel Subasic.
Boudebouz was also involved in Montpellier's second goal when his curling free kick struck Monaco striker Guido Carrillo on the back and settled in the bottom corner of the net.
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Coentrao pulled a goal back for the visitors in the 56th with a low shot past Ligali following a corner before Thomas Lemar levelled nine minutes later after Nabil Dirar's cross.
The stage was then set for Fabinho to step up and score the winning penalty after the hapless Ligali had spilled a cross and bundled into Coentrao as he tried to collect the ball.
(Writin by Ken Ferris; Editing by Pritha Sarkar)
