Pellegrino made six changes to his side, which suffered an embarrassing defeat to Leicester on Wednesday, prompting the manager to play only those 100 per cent committed to the club.
Southampton now sits just four points above the relegation zone and questions of their safety will begin to creep in.
Chelsea dug deep, however, to dispatch Southampton behind Marcos Alonso’s free-kick on the stroke of half-time and came close to adding a second on numerous occasions.
Willian nearly made it a bright start for Chelsea after just a quarter hour when he played a one-two with Alonso, but the Brazilian slipped his shot wide of the far post as he cut across goal.
Fraser Forster was busy early and stayed busy throughout denying Gary Cahill twice in quick succession.
The Southampton keeper then kept out N’Golo Kante’s booming effort, before halting a driving shot from Alonso at his near post all within 33 minutes.
Pedro came close to an opener when he beat Forster, but the deflected attempt struck the post.
Alonso, however, found the breakthrough on his 100th league appearance, as his free-kick from 20-yards snuck past Forster for pace deep in first-half stoppage time.
Southampton had little reaction after the break until Charlie Austin replaced the out-of-form Manolo Gabbiadini.
Austin stepped off the bench and nearly found Saints their undeserved equaliser on 62 minutes, but Thibaut Courtois was there to make the save in one of his few causes for concern.
It was another substitute who came close to an instant impact in the 69th minute when Cesc Fabregas set up Eden Hazard, but the Spaniard was adjudicated to be offside in the build up.
Forster would twice prevent Alonso for adding to his earlier tally and halted Alvaro Morata at the near post seconds a part.
Chelsea: Courtois, Azpilicueta, Christensen, Cahill (c), Moses (Zappacosta 82), Kante, Bakayoko, Alonso, Willian, Hazard (Morata 74), Pedro (Fabregas 68).
Unused subs: Caballero (GK), Rudiger, Drinkwater, Batshuayi.
Goals: Alonso (45+2).
Bookings: Alonso (90+4).
Southampton: Forster, Bertrand, Hoedt, Yoshida (c), Stephens, Cedric (Lemina 16), Romeu, Hojbjerg (Boufal 72), Ward-Prowse, Redmond, Gabbiadini (Austin 61).
Unused subs: McCarthy (GK), van Dijk, Davis, Lemina, Boufal, Tadic.
Bookings: Yoshida (45+2), Redmond (87), Stephens (87).