Rotherham equalised three minutes later when Winnard's attempt to head the ball clear beat his own goalkeeper and fell to Nardiello, who tapped in from six yards.
Accrington midfielder Rommy Boco had a shot cleared off the line towards the end of the first half but both sides went in level at the break.
Three minutes into the second half Nardiello converted Liam Ridehalgh's cross to put the visitors in front, with Dunbavin unable to keep the ball out.
The goal is Nardiello's 10th of the season and his fifth in four appearances.
Rotherham's victory moves them up to fourth in League Two.