Valenciennes looked set to move out of the relegation zone when Ghana striker Majeed Waris put them 2-1 ahead from the penalty spot after Benjamin Andre had cancelled out his earlier strike.
However, there was a dramatic twist in the tale as 18-year-old attacker Baradji - a product of the club's youth system - came off the bench to turn the game on its head in the final minutes.
First, the teenager levelled matters with a fine strike past Valenciennes goalkeeper Magno Novaes in the 87th minute and then he sealed a remarkable victory three minutes later by heading home Paul Lasne's right-wing cross.
Tunisian forward Wissam Ben Yedder had put Toulouse in front after 56 minutes, only for Clarck N'Sikulu to net an 89th-minute equaliser.
Rennes are now six points above the relegation zone after dealing another blow to Marseille's fading European hopes with a 1-0 win at the Stade Velodrome.
The visitors scored the winner 14 minutes from time as midfielder Abdoulaye Doucoure confidently finished off an excellent Rennes counter-attack that involved Jonathan Pitroipa, Jean Makoun and Nelson Oliveira.
Nantes won their first league game in 10 attempts as Michel Der Zakarian's men overcame nine-man Montpellier 2-1 in an ill-tempered affair.
Nantes surged ahead thanks to goals from Serge Gakpe and Koffy Djidji, who then put through his own net to hand Montpellier a lifeline, which was taken away when Joris Marveuax was dismissed in first-half injury time for a rash challenge on Lucas Deaux.
Elsewhere, Bastia held on for a 2-0 win over Reims despite the dismissal of Sambou Yatabare, while Alexy Bosetti's equaliser secured a 1-1 draw for Nice against a Bordeaux team that had Abdou Traore sent off.
