George Byers headed Swansea in front on six minutes, but Mason Bennett quickly equalised before Aaron Lewis restored the visitors' advantage in the 17th minute.
Courtney Baker-Richardson looked to have secured the Swans's fifth successive league win when he put them 3-1 up midway through the second half.
But Luke Thomas's deflected shot gave Derby hope on 77 minutes and within 120 seconds Adam King's own goal rescued a point to move the Rams six points clear of the relegation zone in eighth place.
Swansea were denied a fifth successive victory, but they are four points ahead of Derby in fifth.
Derby: 22 Jan v Birmingham (H) PL Cup
Swansea: 22 Jan v Burnley (H) PL Cup