Goals from Jordan Rhodes and Scott Dann were enough to give Rovers their first win in 10 league games and end a run of two successive defeats.
Rhodes converted an early penalty and Dann headed in his fourth league goal of the season to inflict a second consecutive defeat on the Rams.
Dann notched in the third minute of time added on, even though only one minute was indicated.
Clough said: "We certainly paid for our failures in both penalty areas. We spent the last couple of days chatting about keeping Blackburn quiet. To give the penalty away how we did was amateurish. It gave Blackburn a big lift and then we create three as good chances as you will get away from home.
"Then we conceded the killer second in disputed added time I have to say. He put a minute up and it was two minutes 25 seconds when they added the second which we just wondered what happened. We didn't see a stoppage."
Clough also felt his side should have had a penalty late on after substitute Ben Davies was upended by Morten Gamst Pedersen.
"Even if the penalty had have been given on Ben Davies, which was pretty blatant, then you never know. It would have created a nervy situation for Blackburn," Clough said.