Eustace overhauled his side for what felt like a must win game for the head coach, making six changes from the side that was beaten 4-0 at home by Norwich on Tuesday.
Former Brighton centre-back Shane Duffy was among them as he made his Rams debut a week after joining the club.
And while the presence of the 34-year-old was a welcome one for a leaky defence that has been decimated by injury, the backline repeatedly failed to deal with the energetic presence of Gaizka Larrazabal on the right in the first half.
A superb bit of control and pulled-back pass from the Spaniard should have set Xavier Simons up for an early opener, but the midfielder wastefully blasted his attempt from 15 yards over the bar.
It was after Zetterstrom kept a Chris Forino header out at the back post that Larrazabal again had the Rams reeling - this time his incisive cut-back pass was slotted home first time by Rodrigues.
While the setback prompted half-time jeers, they were replaced by wild celebrations immediately after the interval as Nyambe popped up at the back post to tap home his first ever goal - ending a 312-game wait for the 28-year-old.
The leveller came after excellent work from Carlton Morris and an audacious attempted flicked finish from Mo Fuseini, that eventually rolled into Nyambe's path.
Morris went on to tee up a second within minutes, with his header across goal glanced home by defender Robinson.
Zetterstrom went on to make two huge saves to ensure the Rams kept that lead, denying Larrazabal and Sam Dalby. Sandwiched between those stops was an equally impressive bit of goalkeeping from Jack Bonham at the other end as he denied Fuseini from getting Derby's third in the 54th minute.
It was not until the 94th minute that the Rams made it 3-1, with Brewster pinching the ball from Larrazabal before racing away to score.