Lyon stays third in the table, level on 60 points with Paris Saint-Germain but behind on goal difference.
Lille leads the way on 62 points.
Mathieu Cafaro gave Reims the lead after 33 minutes with a sharp half-volley from a cross by Thomas Foket.
Reims was indebted to goalkeeper Predrag Rajkovic who pulled off three memorable saves in the second half.
Lyon's pressure paid off in the second minute of stoppage-time when Kadewere headed in a cross from Memphis Depay for his 10th league goal of the season.
Lyon and PSG clash next weekend in a key top of the table clash.