Despite appearing to be in the ascendancy early on, Chelsea's lack of craft meant Southampton were comfortable, and that encouragement ultimately led to the home side breaking the deadlock. It was 2-0 on the hour mark, some suspect defending allowing Jan Bednarek to turn home from James Ward-Prowse's free-kick.
The victory kept alive fifth-place Chelsea's slim hopes of qualifying for next season's Champions League, with seven points now separating Chelsea from Tottenham and Liverpool above them.
The visitors were level, four minutes after grabbing their first goal, as Hazard fired home from inside the box.
James Ward-Prowse nearly doubled Southampton's lead on 38 minutes following a well struck 28-yard strike which Thibaut Courtois parried away. However, it was a nasty tackle from Marcos Alonso that stole the half-time headlines as he possibly should've seen red for stamping on the calf of Shane Long.
The striker had positioned himself perfectly between Southampton's center-backs to nod Alonso's delivery into the near corner beyond a helpless Alex McCarthy in the 70th minute. Ward-Prowse swings in a free kick from 35 yards out, on the right. It lands at Giroud's feet and the Frenchman half-volleys it home!
Two minutes, a deep Willian corner to the far post nearly caught McCarthey out as the ball seemingly arrowed into the net, but he managed to get a left hand to the ball and scoop it away from danger at the last moment. Southampton's heads had dropped, and they would drop even lower three minutes later as Chelsea completed their comeback in the 78th minute.
Bednarek's goal was his first in the top flight in only his third game for the club.
Conte admits their performance in the opening 45 minutes was not good enough, but was delighted with the response he got after the break, as the west Londoners successfully overturned a two-goal deficit for the first time in the Premier League since 2002. At least Southampton, playing the same night as ourselves, have no luxury with regards to team rotation, as they are desperately and forlornly trying to maintain their Premier League status.
The Nigeria global was thrown into the game in the 86th in place of Eden Hazard at a time when both teams were cautiously playing to see off regulation time.
Salah's goal, his English Premier League-leading 30th, increased his advantage over Harry Kane in the Golden Boot race while strengthening Liverpool's grip on a top-four finish.
Subs: Moses 6, Pedro 6, Giroud 8.