But Cesc Fabregas slotted home four minutes before half-time for what proved to be the victor.
The Rams nearly forced the game into penalties when David Nugent, who came on as a second-half substitute, thought he had scored but saw his shot rebound off the inside of the post.
Chelsea have been drawn at home to Bournemouth in a repeat of last season's quarter-final.
The lead was short lived as Mount laid on Martyn Waghorn's leveller after 27 minutes before Fabregas' victor.
At some point, this fixture felt more like a celebration than just a cup tie with the number of extravagant storylines and nearly comical subplots that had unfolded before our eyes at Stamford Bridge.
Mount and Tomori are both on loan from Chelsea and Zola enjoyed the duo's performances.More news: Djokovic hails his return to the top as 'phenomenal'
That's my goal. I want to come back and play for Chelsea, but at this moment I'm at Derby and I'm loving it, so that's where my focus is.
There was another back and forth shortly after with Martyn Waghorn atoning for another own goal, this time from Tomori's centre-back partner Richard Keogh.
All five goals were scored before half-time and the Rams' striker David Nugent hit the post in the dying stages of the game, as Derby pushed hard for an equaliser. We need to stamp our authority in that and kick on.
Chelsea's loanees were involved in goals at both ends.
27' - GOAL! Chelsea 2-2 Derby: They have only gone and equalised again! This required Derby to keep their midfielders back to prevent Chelsea from bringing the ball and overloading players in against Derby's defence.
Davide Zappacosta's cross deflected in off Tomori's standing leg as Chelsea took the lead.More news: Gerrard takes full responsibility for Rangers semi-final loss to Aberdeen
Former Chelsea FC forward Zola, who is now working as Maurizio Sarri's assistant manager at Stamford Bridge, is confident Hazard is nearing a return to first-team action.
"He is looking quite good for the weekend [the home game with Crystal Palace on Sunday]".
The Rams stunned Premier League side Manchester United at Old Trafford in the third round, having already overcome Oldham Athletic and Hull City, while Chelsea beat Liverpool last time out.
He said he was "a lucky man to have a night like that" at full-time and was visibly emotional as he thanked supporters. "[It wasn't] great for Fik in the first four minutes", Lampard said after the game.
"I just hope and pray that they can go there and I think as well it's an opportunity for Chelsea to look at their two players under their own lights and see how they are getting on against top, top opposition".
DF Davide Zappacosta, 6 - Provided two assists with unsafe crosses, though it's hard to know how much credit to give him when they both found Derby players.More news: Joel Embiid kept shouting ‘you can’t guard me’ at Andre Drummond