The Saints entered the match in 18th position in the table, and having not won a league game on home soil in 2018.
The result means that Southampton remain inside the Premier League relegation zone, while Chelsea stay fifth in the division - six points clear of sixth-placed Arsenal.
Chelsea boss Antonio Conte made two changes to the team that started the 1-1 draw with West Ham United last weekend, with Andreas Christensen and Davide Zappacosta coming in for Antonio Rudiger and Victor Moses.
As for Southampton, there was a Premier League debut for centre-back Bednarek as Jack Stephens started his three-match suspension.
Neither side had a chance of note in the first 15 minutes of action, but Southampton did have plenty of the ball, with Tadic the player attempting to make things happen in the final third of the field.
Chelsea, meanwhile, were finding it difficult to carve out openings despite the best efforts of Willian, who delivered a couple of dangerous balls into the home side's penalty box.
Southampton were having joy in the wide positions, and it was from that avenue that they managed to make the breakthrough in the 21st minute, with Ryan Bertrand showing excellent pace and strength to drive down the left - holding off Cesar Azpilicueta in the process - before delivering a low cross in to Tadic, who finished into the back of the net.
Chelsea had the opportunity to hit back just before the 30-minute mark when Zappacosta's header found Alvaro Morata inside the Southampton box, but the Spain international nodded over the crossbar.
It was also Southampton that had the final chance of the first period when Ward-Prowse's dangerous free kick fell for Maya Yoshida, but the centre-back could not direct his header towards goal.
Morata curled high and wide of the Southampton post early in the second half as Chelsea looked to respond, before Courtois did well to keep out a close-range Yoshida effort down the other end of the field.
Long's pace was causing Chelsea problems, and Courtois had to save an effort from the Republic of Ireland international in the 54th minute after Christensen had been turned outside his own box.
Southampton goalkeeper Alex McCarthy was busy down the other end, meanwhile, with the 28-year-old keeping out strikes from Hazard and Willian in quick succession.
The home side doubled their advantage on the hour when debutant Bednarek side-footed a wonderful Ward-Prowse free kick into the far corner of the net.
Chelsea were back in the contest in the 70th minute, however, when Giroud, who had not long been on the field, headed a Marcos Alonso cross into the bottom corner of the net.
Hazard then smashed a Willian cross into the roof of the net in the 75th minute as the away side levelled the scores.
Chelsea were not finished though, and scored their third three minutes later when Giroud stroked a loose ball into the bottom corner to break Southampton hearts.
The home side had a late half-chance through Josh Sims, but the midfielder fired straight at Courtois as Chelsea secured all the points in spectacular fashion ahead of their meeting in the semi-finals of the FA Cup next weekend.
SOUTHAMPTON (3-4-2-1): McCarthy; Yoshida, Hoedt, Bednarek (Gabbiadini 92'); Cedric, Romeu, Hojbjerg, Bertrand; Ward-Prowse (Austin 80'), Tadic (Sims 83'); Long
CHELSEA (4-3-3): Courtois; Azpilicueta, Christensen, Cahill; Zappacosta (Pedro 61'), Kante, Fabregas, Alonso; Willian, Morata (Giroud 61'), Hazard (Moses 86')