The left-back secured a dramatic victory for the Stade Louis II club, who remain in the Ligue 1 title race
An early header from Valere Germain gave Claudio Ranieri’s side the early edge but after a lively start they failed to maintain their momentum.
Monaco were punished early in the second half when Oniangue spun brilliantly on the edge of the box to level and though Jeremy Toulalan replied with his first Ligue 1 goal in nearly a decade, the midfielder would equalise once more after a goalkeeping error.
And Kurzawa was on hand to snatch a somewhat undeserved winner five minutes into stoppage time after Aissa Mandi had been sent off.
A surprisingly open start to the match favoured the hosts against opponents they had failed to beat in their two previous meetings this term. In the first five minutes Toulalan volleyed wide following a Joao Moutinho corner but on the next occasion the Portuguese delivered Germain was able to glance a header into the net at the near post.
After such a positive start, Monaco allowed the pace to drop and the match to become a scrappy affair and by the time Reims drew level at the beginning of the second period, it was hard to deny the Champagne club deserved parity.
Odair Fortes burst forcefully away down the right flank before passing inside to Oniangue, who spun Ricardo Carvalho beautifully before holding off a challenge from Eric Abidal and slotting a calm finish past Subastic.
Monaco were awoken by this blow yet when they moved into the lead, the goal was scored by the unlikeliest of figures. Toulalan had scored his only previous Ligue 1 goal against Strasbourg in October 2004 yet a fizzing shot from the edge of the area doubled his career tally and gave the hosts what was to be a brief advantage.
Barely seven minutes later, Reims were presented with a simple opportunity to fight back once more. A free kick into the box was poorly dealt with by Subastic and after the goalkeeper failed to make any contact, the ball broke kindly for Oniangue, who fired into the unguarded net.
Efforts from James Rodriguez and Germain in the closing stages offered some threat to the visiting side, who would finish with 10 men as Aissa Mandi was dismissed for a high, late challenge.
Kurzawa was the victim of that tackle and he made Reims pay just as it seemed they had held on for a point. A James shot bounced off the post five minutes into stoppage time and he reacted quickest to tap home from close range.
Ranieri’s men have closed the gap on Paris Saint-Germain at the summit of the league to just two points prior to the capital side's trip to Toulouse on Sunday, while inflicting just a third away defeat of the season upon Reims.