It was a deserved victory but the Rams had to work hard for their first home league win since December.
George Burley's team dominated the first half without creating a clear opening and it needed a well-worked goal from Smith to break Leeds' resistance.
The visitors had their moment at the start of the second half and almost went in front when Derby substitute Michael Johnson came close to slicing a clearance into his own net.
But the home side finally made their superiority tell when Smith played the ball out to Jeff Kenna on the right and his first-time pass found Smith, who showed good control and composure to go past Neil Sullivan and score from a tight angle.
Smith had been a threat to Leeds in the first half when his tireless running threatened to open up the visitors defence and he brought a good save from Sullivan near the end of the period.
The keeper also did well to keep out a low shot from Ian Taylor and Derby should have done better in stoppage time when Marco Reich got clear on the right but his low cross was cleared.
Derby looked uncertain in the early stages of the second half and when Pablo Mills gave the ball away, David Healy almost scored with a shot that Lee Camp saved with his legs.
But the Smith goal gave Derby a cushion and they extended their lead in the final minute with Leeds pressing for an equaliser.
Derby broke out with Inigo Idiakez playing the ball through to substitute Bolder, whose shot from the edge of the area deflected of Frazer Richardson to leave Sullivan stranded.
The victory lifts Derby into the play-offs places for the first time in two seasons but it could have come at a cost.
Left-back Richard Jackson was taken off after 17 minutes and could now miss Saturday's FA Cup fourth round tie against Fulham.