Derby, who are coached by Chelsea legend Lampard, went behind when Tomori inadvertently deflected a cross into his own net.
Derby found themselves behind again before half-time, as Cesc Fabregas netted much to the annoyance of the visitors who felt there was a foul in the build-up.
"Talking to the Chelsea players and some of their staff after the game, they were all praising us and saying how good we were. The aim is to come up and play in the Premier League".
Blues assistant manager Gianfranco Zola told Sky Sports Hazard is "looking good" to return to action on Sunday against Crystal Palace, clearly a massive blow for the Eagles.
Chelsea - who made eight changes for the game but still fielded 11 internationals - were ahead again before the break, but it wasn't without controversy.
According to a UK Sun report, Lampard insists that the 27-year-old's current form under Italian manager, Maurizio Sarri is far better than that of Argentina's Lionel Messi.
The Blues have given Derby permission to field on-loan duo Mason Mount and Fikayo Tomori. The Premier League is a different level.
It was a special night for the manager who returned to Stamford Bridge for the first time tonight as a manager.
While remaining a star for South Korea, which he led to a gold medal - thereby earning exemption from military service - at the Asian Games in September, Son is no longer guaranteed a first-team place at Tottenham because of fierce competition up front.
PM Unveils Sardar Patel's 2900-Crore Statue Of Unity
Gujarat government chief secretary JN Singh said that dance troupes from across India would perform at the inauguration. Former Gujarat chief ministers Suresh Mehta and Shankersinh Vaghela have also come out against the extravaganza.
Ten-man Arsenal were made to sweat by League One Blackpool before eventually emerging 2-1 winners, with goals from Stefan Lichtsteiner and Emile Smith-Rowe.
The Premier League giants moved into the EFL Cup quarter-finals after the tight win at Stamford Bridge on Wednesday.
Chelsea are without Olivier Giroud, while Eden Hazard is not in the matchday squad due to a back problem.
Predicted Derby County XI: Carson; Bogle, Tomori, Keogh, Forsyth; Wilson, Huddlestone; Waghorn, Mount, Lawrence; Nugent.
Chelsea's loanees were involved in goals at both ends.
Key Stat: Ruben Loftus-Cheek has scored twice as many goals in his past two games than he did in his first 35 for Chelsea.
"We have two massive games coming up but we can never relax".
The Championship side, conquerors of Manchester United in the previous round, provided a properly stern test of Chelsea's undefeated team in a thrilling game and were denied parity - and the possibility of another penalty shootout - only when David Nugent's late shot fizzed beyond Willy Caballero and rebounded back to the goalkeeper off the far post.
41' - GOAL! Chelsea 3-2 Derby: For the third time Chelsea lead this evening, but Derby aren't happy.