A 2-0 lead late in the second half, quickly fell apart as Chelsea came from behind to secure a 3-2 victory away from home.
Dusan Tadic and debutant Jan Bednarek put the hosts in command, but Giroud's first league goals for Chelsea either side of Eden Hazard's equaliser cut the gap on Liverpool and Tottenham in the fight for a top-four finish to seven points.
Antonio Conte is not bothered about claims Napoli coach Maurizio Sarri has been approached with a view to replacing him at Chelsea. They still remain all but eliminated from the top four race, but at least they showed signs of a team, in the second half at least, fighting for something substantial.
The France striker then channelled that never-say-die spirit again when insisting the Blues can not admit defeat in their top-four quest, despite still trailing fourth-placed Tottenham by seven points.
Conte admitted his side had delivered a "very disappointing" first-half performance at St Mary's in his post-match interview but added his praise for his players' response. They have to play for the club and the fans, to show great character like today.
The Blues put up an incredible fightback against the Saints to salvage a victory after going two goals in the first half.
Chelsea defender Antonio Rudiger was axed from Chelsea's match day squad for their trip to Southampton on Saturday for a breaching a golden rule. If you don't have something in your heart, it's very hard to win the game, but today I'm very happy with the second half, and very disappointed with the first.
Magnificent. Again. A masterful midfield performance by Kante, who was dominant throughout, standing out both when Chelsea were bad and when they were on top.
I said in the second half that we must be pleased because I saw the right fire and desire in my players.
The burly Frenchman would reduce the deficit with 20 minutes left to play and Eden Hazard would level proceedings just five minutes later.
"We know we are playing for Chelsea, when you are playing for Chelsea, you need to give everything to the end of the season, that's it. When you are not able to do this, it means you have to take responsibility".
Olivier Giroud has no regrets about leaving Arsenal as he says he has joined the best team in England in Chelsea.
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