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TV replays showed that David Johnson's 19th minute goal should have stood instead of being ruled out for offside.
Johnson appeared to be level when he beat Lee Grant with a rising right- foot shot but Hart was philosophical about the decision.
"David felt that he was on but these things happen," Hart said after the match.
"It was tight and on another day it would stand but I was happy with the team's performance."
That was the only effort Forest managed on target and Derby only tested Darren Ward once when Chris Riggott headed into his arms in the first minute.
The rest of the game was a typical local derby, high on effort but short on quality and there were few real chances.
David Prutton curled a 25-yard shot just wide in the first half and Adam Bolder headed over for Derby.
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A cross from the right was headed down by Izale McLeod but Malcolm Christie blazed the chance over the bar from ten yards.
It was Derby's last opportunity to win the first meeting between the clubs since April 1999 but Forest also had a glimpse of glory in stoppage time.
Marlon Harewood was sent clear by Gareth Williams but his touch let him down and Grant, who was making his full debut, was able to drop on the ball