Chelsea striker Alvaro Morata was again on the bench after a back injury, as Antonio Conte continued with his front three of Hazard, Willian and Pedro.
BBC Sport shared the two lineups:
BBC Sport @BBCSport
Alvaro Morata is back on the bench for Chelsea but the home side stick with their false 9 formation against Southampton. Live 👉 https://t.co/v9TZotj9vO https://t.co/8b0iZszMtv2017-12-16 14:37:02
The first half was a tight affair, although the hosts had the better of the opportunities. Willian was the first to go close, swapping passes with Alonso before firing just wide after 14 minutes.
Southampton goalkeeper Fraser Forster then saved a speculative effort from Tiemoue Bakayoko before tipping an Alonso effort around the post.
Sports reporter Matt McGeehan felt the Blues were missing the influence of Cesc Fabregas:
Matt McGeehan @mattmcgeehan
I think Chelsea need Fabregas in a game like this. Too much tika-taka - need some incisive passing. Yes, Forster has made saves, but most have been regulation2017-12-16 15:34:51
With just two minutes of the first half remaining, Pedro cut inside from the left and saw his deflected shot cannon off the post.
However, Chelsea were not to be denied and eventually broke the deadlock in first-half stoppage time.
Hazard won a free-kick on the edge of the box, and Alonso curled it around the wall and beyond Forster.
Football writer Seb Stafford-Bloor felt goalkeeper Forster should have done better:
Seb Stafford-Bloor @SebSB
The angle meant that the other side of the goal wasn’t in play for Alonso, so Forster probably should be getting to that.2017-12-16 15:54:20
Goal's Nizaar Kinsella illustrated just how dominant Chelsea had been in the first 45 minutes:
Nizaar Kinsella @NizaarKinsella
Drinkwater and Courtois warming up at HT. That's how dominant Chelsea have been, their goalkeeper needs to keep warm with Caballero ahead of the second half. Courtois had 13 touches and made 0 saves in that half. #CHESOU2017-12-16 16:00:33
Southampton struggled to trouble Chelsea after the break and sent on in-form striker Charlie Austin to increase their attacking threat.
The striker almost scored with his first touch, after Nathan Redmond played him in, but his effort was tipped around the post by goalkeeper Thibaut Courtois.
Chelsea responded immediately to that warning as Hazard and Fabregas combined in the box, with the Belgian slotting home only for the offside flag to deny him.
Conte then sent Morata on for Hazard, and the substitute was quickly into the thick of the action, but Forster blocked his close-range shot.
With Chelsea having been unable to make the game safe, Southampton began to press in the closing stages, with the influential Austin testing Courtois once again.
Journalist Dan Levene said it made for a tense finish at Stamford Bridge:
Dan Levene @danlevene
Austin so close to levelling it, as Southampton step things up a gear. Potentially an anxious last five minutes for Blues.2017-12-16 16:43:04
Conte's side managed to hang on for all three points and a narrow victory. The hosts may feel they could have won by more, but some poor finishing let them down, and they endured some nervy moments particularly after Austin's introduction.