The home team's goal came from a penalty by Chris Wood after Aston Villa's Lucas Digne handled the ball when he raised his arm in the penalty area.
Victor Nilsson Lindelöf remained on the bench throughout the match for Aston Villa.
In the Premier League, Nottingham Forest are in 16th place, five points ahead of Tottenham, who are in the relegation zone. Aston Villa, on the other hand, are in fifth place and are very much in the fight for a Champions League place.
The return leg will be played next Thursday at Villa Park.
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