After Alvaro Morata had given the 'Blues' a first half lead, the Englishman and Brazlian put the game beyond doubt with two spectacular strikes.
Barkley was the first to get on the score sheet, he added to his previous assist by picking up a pass from N'golo Kante and powering a low drive past Joe Hart, via the slightest of deflections.
Willian repeated the trick minutes later. He jinked in from the left after being freed by some quick build up and curled and effort into the same corner.
The goals look to have killed off any challenge from a spirited Burnley side.