They showed one change from the side beaten at Charlton with Jason Scotland replacing Guillem Bauza upfront.
Forest almost opened up the scoring in the fourth minute when Robert Earnshaw slipped a pass out wide to Lee Martin and his low cross was fumbled by Swans keeper Dorus de Vries in front of goal, but Ashley Williams hacked the loose ball clear.
It was the Swans who broke the deadlock in the seventh minute when Scotland put Leon Britton in the clear.
He made space before rifling a shot that hit the base of the post before rebounding off the unfortunate Smith in the Forest goal and rolling slowly over the line.
The Swans went close to adding to their lead through Tom Butler and Darren Pratley as their neat passing game threatened to over-run the visitors.
On the half-hour, a lofted pass from Scotland was headed inches wide by Bodde.
Totally against the run of play, Forest moved the ball down to the right corner where Martin crossed, de Vries failed to gather and James Perch was on hand to stab home the equaliser.
Forest began the second half on top and Earnshaw went close in the 53rd minute and then almost broke clear two minutes later only to be blocked by Williams.
Butler was brought down in the area by Wes Morgan in the 70th minute and, although Bodde's penalty was saved by Smith, the Dutchman slotted home the rebound to put the Swans ahead.
They made the game safe in the 86th minute when a cross from Mark Gower was brilliantly flicked into the net by the debut-making Pintado.