It was the 21-year-old Germany international’s first goals for the club after arriving from Bayer Leverkusen for £71m earlier this month.
Tammy Abraham put the Blues ahead and had a role in all of Havertz’s goals.
Ross Barkley and Olivier Giroud were also on target as the hosts eased into the fourth round at Stamford Bridge.
“I’m delighted with Kai,” said Lampard.
“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’s hat-trick was the first of his senior club career and he has now been involved in 26 goals in all competitions in 2020, scoring 18 and making eight assists.
Only Robert Lewandowski (35), Lionel Messi (32) and Cristiano Ronaldo (31) have been involved in more goals among players in the top five European leagues.
“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.”
The Blues will visit either Tottenham or Leyton Orient in the next round.