That lead lasted just five minutes as Hazard levelled to keep Chelsea's two-point lead intact, with Atletico Madrid held to a goalless draw by FK Qarabag in the other Group C game.
Defeat means Southampton stay fifth, while the three points take United, who move into the top half, up to seventh, five points behind league leaders Liverpool with a game in hand.
Mohamed Salah scored twice as Liverpool eventually broke down resilient Newcastle United to open up a three-point lead at the top of the Premier League.
The Portuguese international became the club's all-time top goalscorer with an amazing haul on Saturday, scoring a hatful and also providing an assist in the rout Cristiano Ronaldo scored a sensational five goals and became Real Madrid's record
Jurgen Klopp's side went into this round of fixtures five points clear at the top and, with Arsenal and Manchester City winning their games against Nottingham Forest and Burnley respectively, the Anfield club needed to do likewise against the
Mauricio Pochettino's side could not muster a shot until the 45th minute as Palace tried to frustrate their opponents and pick up a much-needed win to widen the five-point gap to the relegation zone.
Liverpool extended their lead at the top of the Premier League to five points as an impressive second-half display overpowered Bournemouth at the Vitality Stadium.
The outstanding Kante, who was also instrumental in the opening goal, fed substitute Christian Pulisic and his cross was turned in by Mount with five minutes left.
The win means Liverpool cemented their place at the top of Group E, on nine points after three matches, while Toulouse dropped to third, five points off the Reds.
Liverpool squandered the opportunity to return to the top of the Premier League as they were held to a draw in another Old Trafford thriller with Manchester United.
The Madrid coach backed Ronaldo as the "clear" front-runner for the Ballon d'Or this week despite a run of one goal in his last five league games, and he led the way in Vitoria as the pacesetters came from behind to win comfortably.
Kylian Mbappe was on target as Paris St-Germain went top of Group F with a clinical win over AC Milan in the Champions League. The hosts are now two points above Borussia Dortmund, who beat Newcastle 1-0 on Wednesday, in second.