Havertz, who signed with Chelsea from Bayer Leverkusen for a reported £62 million this summer, scored goals in the 28th, 55th and 65th minute to complete the milestone, while Tammy Abraham, Ross Barkley and Olivier Giroud rounded out the Blues rout.
The lopsided result sets up a clash in the next round with either Tottenham or Leyton Orient, who had their third-round match postponed on Tuesday due to positive COVID-19 tests.
Abraham struck first for the hosts in the 19th minute, intercepting a careless back pass to a defender before slotting home from inside the box.
Havertz got his first for his new club nine minutes later with a deft finish from close range after a wonderful Abraham dummy played him in with only the keeper to beat.
Barkley made his presence felt in the 49th minute after more slack Barnsley ball-handling at the back, pouncing and firing past keeper Bradley Collins after a Havertz tackle outside the box played the ball into his path.
Havertz then completed his hat trick in a 10-minute flurry, the first coming from a lovely Abraham layoff and the second from a pass behind the defense that he played between Collins' legs before rolling into an empty net.
Giroud converted a diving header to put the finishing touch on a comprehensive victory at Stamford Bridge.
"I'm delighted with Kai," said Chelsea manager Frank Lampard about the player's performance.
"It was everything I wanted the night to be for him. Because Kai has had no pre-season and signing for the club took a few days out for him as well."
Havertz was one of several notable signings Chelsea made over the summer, bringing the German midfielder from Bayer Leverkusen.
"Tonight was a great exercise for him to see how we want to work off the ball, for him to be a part of that," added Lampard.
"He made a great recovery tackle for Ross' goal, and then to also just have some freedom on the ball and roam in more central areas.
"I thought it was a great combination with Tammy and I was pleased with the way he grew into the game in terms of confidence. It was a great night for Kai, and the first of many for him."