The left-back, unmarked after a late red card for the visitors' Algerian Aissa Mandi for a dangerous tackle, scored from close range after a rebound off the post in 95th minute.
PSG visit 10th-placed Toulouse on Sunday looking to restore a five-point lead over Monaco, who are 11 points clear of the chasing pack.
The principality side went ahead when Valere Germain's pinpoint header from a Joao Moutinho corner flashed into the net in the eighth minute and Monaco looked in control of the game.
But eight minutes into the second half Oniangue caught the home side by surprise, muscling past experienced defenders Ricardo Carvalho and Eric Abidal to level the scores.
Monaco, cheered on by a sparse crowd at the Stade Louis II, hit back 10 minutes later when 30-year-old Jeremy Toulalan took the ball on the wing, stepped inside and drilled it low into the net from just outside the area.
Oniangue then took advantage of calamitous goalkeeping in the 71st minute when he smashed home a loose ball after Danijel Subasic rushed off his line but failed to collect a free kick.
Reims battled to keep Monaco out in the closing minutes but, with Mandi sent off in the 83rd, they lost their shape.
Monaco's new Bulgarian striker Dimitar Berbatov, just back from injury, was introduced for the closing minutes, but it was defender Kurzawa who struck the winning blow.
- Sports & Recreation
- Prince Oniangue
- Claudio Ranieri
- Joao Moutinho