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Monaco are now second in the standings with 20 points from 11 games, four behind PSG, who host Leipzig in Group H on Tuesday trailing the German side and Manchester United by three points.
Centre-back Abdou Diallo, who fouled Volland for Monaco's penalty, was sent off.
Mbappe, back with PSG since picking up a thigh injury against Nantes on Oct. 31, struck first in the 25th minute with a shot into the top corner.
Moise Kean was denied PSG's third by the Video Assistant Referee (VAR) for being in an offside position two minutes later.
It was referee Clement Turpin who then denied Mbappe his third goal for being offside one minute before the break.
Monaco raised their level after the interval and Volland reduced the arrears in the 52nd minute with a shot from close range.
The momentum had swung and Volland levelled in the 66th, again from close range from Fabregas's cross.
The Spanish midfielder then coolly converted the winning penalty.