The Blues proved that they could cope without injured talisman Eden Hazard as four different scorers stepped up to send them above Manchester City into second in the table.
One poor clearing punch from the Chelsea goalkeeper could have proved costly on 17 minutes when the ball fell to Robbie Brady, who was not far away from marking his first appearance in 10 months with a goal when his skimming volley whistled by the post.
At the other end, Joe Hart looked more assured as he showed sharp reactions to tip Alvaro Morata's header over the crossbar; a skilful effort from the Spaniard as he diverted Barkley's wayward shot towards the top corner.
In the absence of Hazard, Willian was Chelsea's go-to man and he relished the responsibility, and the space afforded to him on the left flank, striking the base of the post on 21 minutes following a through-ball from Morata.
The woodwork spared Burnley only for a minute as the Spaniard put the visitors into a 22nd-minute lead with the help of a perfectly-weighted pass from Barkley, poking the ball beyond Hart for his third goal in his last five matches.
Burnley spurned a decent chance to immediately draw level as James Tarkowski headed over from a corner but they otherwise spent the remainder of the first half on the back foot.
Pedro, who replaced Hazard in Chelsea's starting lineup, limped off after half an hour, allowing Ruben Loftus-Cheek the chance to stake his claim for more Premier League minutes, having again found himself on the bench after his European hat-trick.
Morata had a couple of chances to double his tally, and Chelsea's lead, before the half-time whistle, dragging a shot wide from eight yards out after skilfully turning away from Tarkowski, and later latching onto Jorginho's pass with a powerful volley which Hart did well to block.
The former Manchester City goalkeeper could do little to keep out Barkley's 25-yard drive on 57 minutes; a left-footed shot into the bottom corner indicative of the rediscovered confidence that the former Everton midfielder is showing under Maurizio Sarri.
Chelsea soon put the game beyond doubt as Willian cut in from the right and bent a shot into the bottom corner via a slight deflection - the eighth goal that the Clarets, so sound defensively last season, had conceded in their last two Premier League outings.
Substitute Olivier Giroud twice came close to ending his five-month Chelsea drought, whipping a shot narrowly wide from Willian's layoff before seeing Hart push a thumping stoppage-time header onto the crossbar.
There was still time for Loftus-Cheek to cap a fine week for both himself and the Blues with a 93rd-minute fourth, a bit of a scrappy finish from a corner, but Chelsea's performance was otherwise very stylish, even without Hazard.
The Blues welcome former hero Lampard and his Derby County side in the EFL Cup on Wednesday, while Burnley will be glad to see the back of the Premier League's elite as they visit West Ham United, level with them on eight points, next weekend.
BURNLEY (4-5-1): Hart; Lowton, Tarkowski, Mee, Taylor; Gudmundsson, Cork (Westwood, 73'), Hendrick (Barnes, 65'), Defour, Brady; Vokes (Wood, 59')
CHELSEA (4-3-3): Kepa; Azpilicueta, Luiz, Rudiger, Alonso; Jorginho (Fabregas, 78'), Kante, Barkley; Willian, Pedro (Loftus-Cheek, 30'), Morata (Giroud, 74')