The in-form Kevin Volland brilliantly volleyed home his 10th Ligue 1 goal of the season to break the deadlock in the 24th minute, before Ben Yedder stayed onside to head his side's second 11 minutes later.
A penalty from Ben Yedder looked to have sealed a comfortable three points for Monaco, but a spirited comeback from Montpelier set up a thrilling finale.
First 18-year-old Elye Wahi scored his first senior goal for Montpellier to get his side back into the game before Andy Delort added a second five minutes later.
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Delort almost completed a stunning turnaround but was denied by a fantastic stop from Monaco goalkeeper Benjamin Lecomte.
The result handed Monaco their fourth win from their last five league games.
They have 36 points from 20 games, three points adrift of third placed Lille. Lyon top the standings with 40 points. Montpellier stay eighth with 28 points.
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Draxler and Gueye make it eight wins from eight for PSG against Montpellier