Chris Wood converted a second-half penalty to give Vitor Pereira's side a narrow first-leg advantage ahead of the return at Villa Park next Thursday.
The tie's defining controversy came when Elliot Anderson caught Ollie Watkins with his studs in a sliding challenge — a moment that passed without a red card on the night and without VAR intervention.
Aston Villa manager Unai Emery was incensed after the game.
"It's a clear red card. I don't understand why the VAR is not calling the referee because it's so clear. He could break his ankle. VAR — where are you? It is your responsibility."
He was pointed in his assessment of the refereeing overall, with the sarcasm barely concealed.
"The referee — fantastic, fantastic job. Ten out of ten."
On the result itself, Emery remained composed.
"We couldn't score, we conceded, but we controlled. We had our chances to score and had momentum. The match is not finished — it continues next week. We must come back."
The penalty that settled the tie came in the 71st minute when Lucas Digne handled Omari Hutchinson's cross, with referee João Pinheiro pointing to the spot after a VAR review. Wood finished emphatically.
Nottingham Forest manager Vitor Pereira paid tribute to a squad that has worked with four different managers this season.
"Their spirit, four managers in a season, and finishing at this level competing together — this is an honour for me."
The second leg takes place at Villa Park on 7 May. The Europa League final is in Istanbul.