Williams had already headed a good chance over when his misplaced back-pass caught goalkeeper Dorus de Vries off guard to hand the Rams a comical seventh-minute lead.
The Wales international was having an afternoon he would not forget anytime soon as he then headed against the crossbar at the right end.
But it was left for Steven Davies to show him how it is done with a header from Conor Doyle's corner shortly after the restart.
Substitute Darren Pratley headed Swansea a 65th-minute lifeline while Stephen Dobbie hit the crossbar, but Derby held firm to clinch only their third win in 19 npower Championship matches and provide a huge boost to their survival ambitions.
Source: PA
Source: PA