Returning Romelu Lukaku struck as Chelsea cruised to a comfortable victory, with Timo Werner, Callum Hudson-Odoi, Andreas Christensen and Hakim Ziyech all also finding the net.
The 91 league places between the two sides told throughout, with Chelsea still able to make seven changes despite a raft of injuries and Covid-19 absentees.
Akwasi Asante tapped in a late Chesterfield consolation to spark jubilant celebrations among their raucous travelling fans.
"I am happy with the attitude with how we started. We started strong and well," said Tuchel. "We decided the match very early and this is what we wanted,
"At 5-0, we had a lot of changes in crucial positions and has some players who lacked some rhythm and confidence in the position they play. The job is done and we decided the match in the first half that was very good.
"It was simply the moment and there was different level of intensity against a fifth division team. We have to share the minutes and make sure there were players that could rest to be in the best shape possible for the upcoming games."
Meanwhile, Cambridge boss Mark Bonner saluted his players as they proved the magic of the FA Cup is still alive after the League One side dumped Premier League Newcastle out of the competition.
Joe Ironside’s second-half goal secured a famous 1-0 third-round win at St James’ Park on an afternoon when recent Magpies signing Kieran Trippier was handed an insight into the problems his new employers – who have lost striker Callum Wilson to injury for eight weeks – face.
Asked about the suggestion that the FA Cup is not what it once was, Bonner said: "It certainly is to teams at our level.
"I understand that some say that and why – there’s more football now than there ever was before.
"But the magic of the FA Cup, I guess, is about the underdog pulling out a surprise story. We knew going into today there’s always an upset somewhere and whether we could be it, we didn’t know.
"Somehow we found a way of doing it and I’m delighted that we have done."