The left back secured a dramatic victory for the Stade Louis II club, which remains in the Ligue 1 title race.
An early header from Valere Germain gave Claudio Ranieri’s side the early edge, but after a bubbly start it failed to maintain its momentum. Monaco was 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 equalize once more after a goalkeeping error. Kurzawa was on hand to snatch a somewhat undeserved winner five minutes into stoppage time after Aissa Mandi has been sent off.
A surprisingly open start to the match favored the hosts against opponents they had failed to beat in their two previous meetings this season. In the first five minutes, Toulalan volleyed wide following a Joao Moutinho corner but on the next occasion the Portuguese delivered Germain, who 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 rather scrappier 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 prove a temporary advantage.
Barely seven minutes later, Reims were presented with a simple opportunity to fight back once more. A free kick into the box was inconclusively 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, which 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 to 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.