Tuchel extended his unbeaten run in charge of Chelsea to 11 games and Havertz put in one of his best performances at Stamford Bridge.
The Germany international was heavily involved on the night. His flick from a Marcos Alonso cross deflected off Everton defender Ben Godfrey and into the back of the net to open the scoring.
In the second half he saw a goal ruled out after he was adjudged to have handballed it in the buildup but he then raced through and won a penalty after being felled by Jordan Pickford.
Jorginho stepped up to slot home and wrap up three points for the Blues, who tightened their grip on a top-four spot.
Tuchel was delighted with the efforts of his players and singled out Havertz, who arrived from Bayer Leverkusen for £70million last summer, for special praise after the win.
‘We deserved to win. It was a very difficult first half but we controlled it completely,’ Tuchel told BT Sport.
‘Second half we increased our level and had a lot of dangerous attacks.
‘Very pleased with Kai. It was the trust we gave him and used the trust we gave him.
‘You see the quality of the players and the club. It is a pleasure to be on the sideline and work with the team.
‘Everybody in the club is doing everything to compete at this level.’