The Blues took the lead on 22 minutes, Barkley playing in Alvaro Morata who found the bottom right corner after Willian had hit the post a minute earlier.
Barkley found that same corner 12 minutes into the second half to double Chelsea's advantage.
Five minutes later it was 3-0 as Willian received the ball from Barkley, glided past Johann Berg Gudmundsson and curled home.
Ruben Loftus-Cheek fired a fourth into the roof of the net in second-half stoppage time.
The result puts Chelsea above Manchester City and into second, with City playing on Monday night, while Burnley slip to 15th with eight points.
Burnley: 3 Nov v West Ham United (A)
Chelsea: 4 Nov v Crystal Palace (H)