Craig Bryson had Derby's first attempt of the match and was unlucky to see his long-range effort miss the target.
After Eric Lichaj had seen a powerful effort deflect over, Forest took the lead form the resulting corner. Defender Jack Hobbs met Andy Reid's precise delivery with a firm header that gave Grant no chance.
Jamie Mackie had a great chance to double the lead for Forest before the break but was denied from point-blank range by Grant.
The home side continued to dominate after the break and on the hour mark Darius Henderson found himself unmarked inside the box but failed to find the target.
Forest were gifted the chance to score their second of the match after Richard Keogh conceded a penalty for a foul on Chris Cohen and received his second booking of the match. Henderson stepped up to take the kick but Grant dived to his right to push the strike wide of the post.
Conor Sammon missed a gilt-edged opportunity to score for Derby when he struck his injury-time effort straight at Darlow, before Reid saw his own strike parried by Grant as Forest held on to claim victory over their Midland rivals.