Tommy Smith hit the post before Derby's Inigo Idiakez smashed in from 30 yards, only for Ade Akinbiyi to level.
Gifton Noel-Williams nodded in Michael Duff's cross to make it 2-1 but Wayne Thomas saw red for elbowing Rasiak.
Rams boss Phil Brown was sent off as the tension increased and it all looked lost for his team until Rasiak headed Smith's cross in off the bar.
Derby boss Brown:
"I complained to the referee and he sent me off because I walked onto the pitch.
"We started with a seven-ball system and it was abandoned. I had to bring it to the attention of the referee.
"When the ball goes into the stand it's not going to come back fast when they are down to 10 men. It wasn't fair."
Burnley: Coyne, Duff, Thomas, McGreal, Harley, Garreth O'Connor, Hyde, James O'Connor, Branch, Akinbiyi, Noel-Williams (Lowe 57).
Subs Not Used: Jensen, McCann, Bermingham, Spicer.
Sent Off: Thomas (55).
Booked: Branch, Duff, Garreth O'Connor.
Goals: Akinbiyi 46, Noel-Williams 53.
Derby: Camp, Edworthy (Seth Johnson 66), Michael Johnson (Tudgay 86), Davies, Jackson, Bisgaard, Bolder (Peschisolido 58), Idiakez, Smith, Rasiak, Thirlwell.
Subs Not Used: Grant, Mills.
Goals: Idiakez 30, Rasiak 90.
Ref: G Laws (Tyne & Wear).