Goals from Dusan Tadic and debutant Jan Bednarek looked to have put Saints in control, but Giroud's introduction on 61 minutes proved decisive.
Brendan Rodgers is firmly in Chelsea's thinking to replace Antonio Conte this summer, with the Blues having drawn up a four-man shortlist, claim reports.
'As long as mathematically it is going to be possible, we are going to believe, ' said Giroud, 'so we take the games one by one and we have maybe five or six finals to play.
Mane opened the scoring on seven minutes, Salah headed in for his 30th League goal of the season and Firmino's excellent finish- his 25th goal of the season in all competition- completed a win which brought the Anfield club to within a point of Man United in second place.
It was Conte's changes that turned the game in Chelsea's favour, with Giroud grabbing his first and second for the club, but the manager was more concerned with the difference in his team as a whole for Chelsea's first league comeback from two goals down since 2002.More news: Facebook Admits It Collects User Data Even From Non-Users
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"As has been the case all year apparently we've got a soft underbelly", said Saints boss Mark Hughes, whose three Premier League games in charge have now all ended in defeats. "For sure, in the first half I was very disappointed for the way we were playing". Here's how we rated each Blues player in the 3-2 comeback victory at St Mary's.
He has managed to only score one goal, which came against Hull City during an FA Cup clash, and he has mainly been used as a backup option to Alvaro Morata and Eden Hazard.
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. However, with the season coming to end, the players need to give it their all to make to close the season for the fans of the club who have been with them through thick and thin.
"I don't know why the desire wasn't there but the coach is the first person to have the fault".
That goal was quickly followed by the equaliser, with Willian's low cross from the left finding Hazard, who was able to take a touch before firing home from inside the area to spare his side's blushes.
"I thought they did that today, and against Arsenal".