The Swans were unchanged following the defeat at Crystal Palace on Saturday and they pinned the Rams in their own half straight from the whistle.
A wayward shot from Nacer Barazite was all the visitors could muster in the opening ten minutes.
The Swansea approach play was neat and tidy, but the final ball to often went astray, while the visitors were playing like a side low on confidence despite bagging their first victory at the weekend.
Totally against the run of play Emanuel Villa beat the offside trap to close in on a Paul Green cross, only to slice the ball well wide of an open goal.
Neither side made changes during the interval and once again the Swans were first on the attack, with a Mark Gower centre blasting across the visitors' danger area with no-one on hand to get the final touch.
On the hour Gorka Pintado replaced Darren Pratley to give Jason Scotland some help up front.
Gower struck the post with Roy Carroll beaten in the 61st minute but Pintado failed to get any power behind the rebound and Carroll raced across to save.
One minute later the Swans defence failed to clear and Pearson cut through to sidefoot beyond Dorus de Vries to give his side the lead.
Pintado headed a Thomas Butler cross over the bar from close in as the Swans pushed for an equaliser and it came in the 76th minute when Williams rammed a header into the net from a Jordi Gomez corner.
Both sides went looking for the winner in the closing stages, but though Carroll was the busier of the keepers, the Swans were unable to add to their score.