It looked as if last season's giantkillers would clinch a comfortable win after storming into a 3-0 lead, but they let their advantage slip when the Rams hit back after they had looked well beaten.
On-loan Jonathon Forte fired Rovers in front in the sixth minute with a cracking left-foot shot after James O'Connor and Lewis Guy had combined to set up the chance.
Derby were way below their best and struggled to respond, although Giles Barnes fired wide in the 16th minute.
But it was Doncaster who scored again in the 28th minute when outstanding on-loan midfielder Brian Stock fired home from just inside the penalty area after Paul Green had teed him up following a free-kick.
Derby were struggling badly at this stage and they had keeper Lee Grant to thank for preventing Forte from adding a third when he dived full length to tip away a fierce low shot.
The visitors came out for the second half in determined mood, but Rovers rocked them again when Forte scored against the run of play after O'Connor and Guy had linked up to slice the visitors' defence apart.
The game looked safe at that stage but Derby hit back when big striker Steve Howard fired home after being set up by Adam Boulder in the 72nd minute.
Five minutes later former Doncaster player Darren Moore gave the visitors a real lifeline when he headed home from close range, with Rovers appealing for a foul after Howard appeared guilty of pushing in the build-up.
Another lapse of concentration by the home side led to Derby snatching a dramatic equaliser in the 88th minute, when on-loan Arsenal striker Arturo Lupoli raced on to a long ball to fire home at the second attempt.
It forced the game into extra-time where Lupoli was sent off for a second yellow card after pulling substitute Anthony Griffiths back in the 95th minute.
Rovers went close to winning the game several times, but were denied by Grant's reflexes as Derby held firm to send the game into a penalty shootout.
Seth Johnson, Howard, Matt Oakley, Boulder, Ryan Smith, Moore and Mo Camara scored with Derby's first seven penalties.
They were matched by Green, Sean Thornton, Jason Price, Sean McDaid, Stock, Gareth Roberts and Adam Lockwood.
Lewin Nyatanga missed Derby's eighth attempt and the home crowd went wild with delight when O'Connor sent Grant the wrong way to win the match with Rovers' eighth successful spot-kick.