However, it was the away side who took the lead after 14 minutes when Nakhi Wells tapped home from close range.
Wells almost made it 2-0 shortly after, but Lee Grant spared the blushes of Keogh, whose underhit pass back to his goalkeeper granted the Huddersfield striker another chance.
The lead lasted little more than 10 minutes, though, as Russell drew Derby level with a composed finish after an incisive through-ball.
On the stroke of half time, Bamford could have put Derby ahead, but the youngster blazed over the bar from close range.
Just five minutes after the break, Smithies inexplicably palmed the ball into his own net from a seemingly innocuous cross by Craig Forsyth.
Huddersfield were shown their second red card of the afternoon when skipper Peter Clarke when shown a second yellow card for bringing down Will Hughes in the box. Chris Martin coolly dispatched the penalty to put Derby into an unassailable 3-1 lead.