Frank Lampard made eight changes to the side that had started the defeat to Liverpool on Sunday, handing Thiago Silva a Blues debut and giving starts to the likes of Abraham and Callum Hudson-Odoi, as well as Willy Caballero in goal.
Hudson-Odoi missed a golden chance to open the scoring inside a minute after being slipped in by Havertz, but Barnsley gave Caballero plenty of early work, too, Cauley Woodrow forcing the Argentine to tip over the bar with a 25-yard curler before Aapo Halme headed straight at him from close range.
On 19 minutes, however, a huge mix-up in the visitors’ backline handed Chelsea the lead, Abraham pouncing to thrash home his first goal of the season.
Mason Mount had already wasted a glorious opportunity to double the lead when his pass was cleverly dummied by Abraham, allowing it to run through to Havertz, who steered coolly into the bottom corner to open his account and make it two.
Still, Barnsley had more chance to reduce the deficit before the interval, Patrick Schmidt, Dominik Frieser and Marcel Ritzmaier all keeping Caballero honest.
The game was as good as over five minutes after the restart, with Havertz again involved, winning the ball back on the edge of the Barnsley box, giving Barkley space to drive forward and fire past Bradley Collins.
Barnsley were once again masters of their own downfall for the fourth goal, this time Mount the man to nick possession and Abraham unselfishly flicked his cross back into the path of Havertz for the German to get his second.
The former Bayer Leverkusen man completed his hat-trick soon after the hour mark after more terrific link-up play from Abraham, poking through Collins’ legs and walking into an empty net.
That was the cue for his withdrawal, Lampard using the chance to hand Ben Chilwell his debut, but there was no let up, with Abraham going close to adding his second from Hudson-Odoi’s low cross, before he was replaced by Giroud.
The two substitutes combined to make it six late on when Giroud stooped to head Chilwell’s cross into the corner and cap a perfect night for Lampard’s men.