Eden Hazard finished smartly to put Chelsea level minutes later, and Giroud would come up with the goods again, scoring the victor off his foot to hand the visiting Londoners a 3-2 win.
He keeps saying "I'm trying my best... this is the situation, we try for the fans" but it's the same tone.
"I said when I came on to Eden Hazard, we need to fight to the end (and) keep believing. Let's do the job and we will see what the other teams do".
Southampton got their reward for a good start after 20 minutes as Ryan Bertrand's blistering run down the left ended with the ball being cut back to Dusan Tadic, who tapped it in past Thibaut Courtois. It was suggested that Chelsea looked like they weren't playing for the manager as they struggled until that point, but Conte denies that was the case.
"I must be honest and I was very angry with our first half because we did not show great desire, great will to fight and will to play in the right way".
While Jose Mourinho's United came from behind to secure a thrilling 3-2 win, the Red Devils need to avoid defeat against lowly West Brom at Old Trafford or the title will head to the Etihad Stadium.
"Our first half was very negative", said Chelsea boss Antonio Conte. If you don't have something in your heart, it's very hard to come back and win.
And their frustration overspilled as tempers flared when Marcos Alonso was lucky to escape unpunished for raking his studs down Shane Long's calf.
Bednarek marked his Premier League debut in style when he half-volleyed expertly across the goal and into the far corner from Ward-Prowse's free-kick on the hour mark.
Chelsea next face a challenging trip to Burnley in the league, while Southampton travel to Leicester before the two sides meet again in an FA Cup semifinal at Wembley on April 22.
Less than 10 minutes later, Conte's men were in the lead through a strike from Hazard and another from Giroud, earning a much-needed three points.
Conte responded by introducing Giroud for Morata and the Frenchman quickly had an impact, heading Alonso's cross in from just inside the area.
He did eventually threaten just before the half-hour mark, heading over from close range, but at that stage Chelsea had already been trailing for six minutes.