However, their luck ran out after half-time as a Chris Martin hat-trick, his first in the Football League, and goals from Craig Bryson and captain Richard Keogh saw the Rams romp back to win handsomely.
They were helped by Blackpool, though, as they had Neal Bishop and substitute Angel Martinez sent off.
It was the Seasiders' fifth red card in a little over 90 minutes of football following their lack of discipline in midweek at Yeovil, where Kirk Broadfoot, Ricardo Fuller and Gary MacKenzie were all sent off in the final five minutes.
It was also an eighth dismissal of the season for Paul Ince's team.
"I don't think it's a lack of discipline," Thompson said.
"It might be in part but it is also down to a lack of judgement.
"It comes down top self control. When you are on a yellow card you are walking as tightrope and the last thing you need to do is make a rash tackle.
"That's what's letting us down at the moment and it has to stop happening.
"It can be down to the discretion of the referee too. The linesman gave the penalty for the first red card but Bishop seemed to come out with ball, which suggests he got the ball.
"We also didn't think he was the last man as it was at an angle where were getting players back in position.
"The lads are just as frustrated as we are. There weren't any bad tackles in the game and yet again we have two sent off.
"We've just got a team that competes very well. If you look at the tackles, they aren't high or with studs showing or career threatening or anything."