A superb late save from Thibaut Courtois denied Sunderland a point against title-hopefuls Chelsea at the Stadium of Light.
Cesc Fabregas had given Antonio Conte's side the lead five minutes before the break and Sunderland were unable to find a way back into the game.
Patrick van Aanholt almost produced an injury time equaliser but Chelsea's keeper made a superb one-handed save to ensure Chelsea left with all three points.
Defeat leaves Sunderland bottom of the Premier League, four points from safety.
Little battler Bradley Lowery was the star of the show before kick-off, taking part in a penalty shoot-out against Chelsea.
The Sunderland fan has captured the hearts of the nation with his brave battle against neuroblastoma and led the side out alongside Jermain Defoe.
Defoe led the Sunderland attack against Chelsea but his usual strike partner Victor Anichebe missed out through injury, one of four changes to the Sunderland side.
John O'Shea, Adnan Januzaj and Fabio Borini the other players to come in with Steven Pienaar out through injury, Seb Larsson on the bench and Didier Ndong suspended.
The opening stages were tight, with Sunderland's new-look back three more than holding their own against table toppers Chelsea.
Cesc Fabregas was dominating the midfield but Jordan Pickford wasn't tested.
Sunderland hadn't challenged Courtois either, Adnan Januzaj blazing an effort wide from the edge of the area.
Ten minutes before half-time, Pickford produced a stunning one-handed save to palm away Pedro's close-range effort.
The offside flag was up but the England Under-21 stopper wasn't to know that.
There was little in the game but Chelsea took the lead five minutes before half-time, Sunderland gave the ball away on the half-way line and the Blues capitalised.
Fabregas played a one-two with Willian before curling home a smart finish from the edge of the penalty area into the corner of the net.
Half-time: Sunderland 0 Chelsea 1
Sunderland should have been level within two minutes of the re-start. Defoe slipped Januzaj in but his low effort was brilliantly saved by Courtois to deny his fellow Belgian.
Chelsea hit the bar, Willian's effort deflected off Djilobodji, with Pickford down low to save from Victor Moses.
Pickford was by far the busier of the two goalkeepers, saving from Costa as well to keep the Chelsea forward line at bay.
Sunderland were then dealt an injury blow in the 59th minute, Billy Jones taken off with Donald Love on in his place.
Pickford again called into action to save from Alonso and Moses again. It wasn't all one way traffic, though, Defoe's clever near post flick going just wide.
With Sunderland only trailing by one goal heading into the final ten minutes, Moyes brought Wahbi Khazri on for Fabio Borini.
Sunderland couldn't find a breakthrough, though, against title-hopefuls Chelsea, Jason Denayer heading straight at Courtois late on from a corner.
The Belgian stopper producing a brilliant one-handed save late on to deny van Aanholt an equaliser.
Full-time: Sunderland 0 Chelsea 1
Sunderland (5-3-2) Pickford; Jones (Love, 59), Van Aanholt, Djilobodji, Kone; O'Shea (C), Denayer, Kirchhoff (Larsson, 57); Januzaj, Borini (Khazri, 82), Defoe
Subs Not Used: Mannone, Asoro, Maja, Honeyman.
Booked: Defoe (43), O'Shea (45), Borini (70)
Chelsea (3-4-3)): Courtois; Azpilicueta, David Luiz, Cahill (C); Moses (Ivanovic, 90), Kante, Fabregas, Alonso; Willian (Chalobah, 88), Diego Costa, Pedro (Matic, 77).
Subs Not Used: Begovic, Zouma, Matic, Loftus-Cheek, Batshuayi.
Booked: Pedro (57), Moses (90)
Goals: Fabregas (40)
Referee: Neil Swarbrick (Preston)