Buchanan's beauty sends Derby County into round two of the Carabao Cup to face Nottingham Forest
The teenager fired home a powerful shot with 12 minutes to go which secured the Rams' passage through after an encounter they'd edged throughout having made several changes to the side which drew at home to Swansea on Saturday.
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Hide AdJack Marriott and Buchanan both went close during the first-half, with United forcing debutant keeper Ben Hamer into a save on 42 minutes from Alex Gilliead's effort and then Jason Knight's shot for the Rams also being saved right on half-time.
Hamer did well to save Adam Hammill's strike early in the second-half before Jamie Paterson then went through one-on-one with United keeper Rory Watson but the home custodian blocked his shot.
Mason Bennett and Kristian Bielik both saw shots saved before the winner came on 78 minutes, Buchanan chesting the ball down before running forward and blasting the ball home.
That would be the last key action as the Rams held on to maintain their lead and ensure their passage through.
The Rams will now travel to face arch rivals Nottingham Forest in the next round.