Enzo Maresca changed his entire starting XI for the visit of the fourth-tier leaders, but it had no effect on the on-field chemistry, as the Blues cut through their opponents at will to rack up a big scoreline.
Nkunku set the ball rolling with yet another cup goal, following his two against Servette in Conference League qualifying, when he rolled past Barrow keeper Paul Farman after a delicately lobbed through ball from Joao Felix.
Inside the opening quarter of an hour, Nkunku had his second, as his fellow Frenchman Malo Gusto crossed low from the right before the striker's fine flick left Farman helpless.
The Barrow stopper's night got even worse on the half-hour mark, when Joao Felix's free kick clattered back off the post, hitting the stranded goalkeeper on the back and rolled over the line for an own goal.
Chelsea came out firing in the second half, and Barrow were cut open with ease once more, with Nkunku feeding Mykhaylo Mudryk down the left, and the Ukrainian looked up and squared for Pedro Neto, who scored his first Chelsea goal.
The chances kept on coming later in the second half for Chelsea, with youth players Josh Acheampong and Tyrique George given game time, and the latter squandered a golden opportunity to make it five with just under 20 minutes remaining.
Chelsea's eventual fifth was an absolute gift, handed on a plate to Nkunku to notch his third, as Farman tried to dribble past the Chelsea forward, but conceded possession and had an empty net to roll the ball into.
The final 15 minutes were a procession, as Chelsea comfortably saw the game out against a demoralised Barrow, putting their name into the hat for the fourth round draw.
CHELSEA VS. BARROW HIGHLIGHTS
Nkunku goal vs. Barrow (8th min, Chelsea 1-0 Barrow)
Chelsea lead in the blink of an eye, after dealing with a few Barrow set-pieces, the hosts show their class, as Felix's wonderful lobbed pass feeds Nkunku, and he remains composed to roll the ball beyond Farman.
Nkunku goal vs. Barrow (15th min, Chelsea 2-0 Barrow)
Another superb finish from Nkunku and the Chelsea man is on a hat-trick inside a quarter of an hour! Gusto's low cross find the striker in the six-yard box, and a nonchalant backheel flick finds it way into the far corner.
Farman own goal vs. Barrow (28th min, Chelsea 3-0 Barrow)
An unfortunate moment for the Barrow keeper, as Felix's 25-yard free kick hits the woodwork, but bounces off his back and rolls agonisingly over the goal line for Chelsea's third inside half an hour.
Pedro Neto goal vs. Barrow (48th min, Chelsea 4-0 Barrow)
Pedro Neto has his first Chelsea goal! A slick move sees Nkunku feed Mudryk down the left, and the wide man got his head up to find an unmarked Neto in the centre of the area, and the Portuguese international had the easiest of finishes.
Nkunku goal vs. Barrow (75th min, Chelsea 5-0 Barrow)
It is a horror show for Farman, who is picking the ball out of his net once again, but even more sheepishly this time, as his awful mistake sees him lose possession to Nkunku in his own area, leaving the Chelsea man with an open goal to knock into to seal his hat-trick.
MAN OF THE MATCH - CHRISTOPHER NKUNKU
Nkunku has not been able to force his way back into the Chelsea after dropping out following the opening day defeat to Manchester City, but he has given Maresca something to think about with today's performance.
The form of Nicolas Jackson will make it tough for the 26-year-old to break into the team up front, but Nkunku is a versatile individual, and scoring a hat-trick against any opposition will never go unnoticed, and he will now have to be considered for Saturday's clash with Brighton & Hove Albion.
CHELSEA VS. BARROW MATCH STATS
Possession: Chelsea 75%-25% Barrow
Shots: Chelsea 21-5 Barrow
Shots on target: Chelsea 11-2 Barrow
Corners: Chelsea 8-4 Barrow
Fouls: Chelsea 16-10 Barrow
BEST STATS
WHAT NEXT?
Chelsea are back at Stamford Bridge on Saturday when they welcome Brighton to West London in the Premier League, while Barrow will look to maintain their position at the top of League Two when they travel to second-placed Gillingham.