Harrad struck on the stroke of half-time to leave Forrest Green's Conference survival hopes hanging by a thread.
The Gloucester side staged a valiant second half fight back but were denied at least a point by a combination of the woodwork and two fine saves by goal keeper Saul Deeney.
Clough admitted: "It was a very, very hard earned three points. The only positives from our point of view were the clean sheet and the goal.
A down cast Forrest Green boss Gary Owers had a mixed response to his sides 11th game without a win.
He said: "In terms of chances created we probably deserved something from the game but I was disappointed that we didn't play with the tempo you need when you are fighting for survival."
Chances were few and far between in a scrappy first half, with a big striker Julian Alsop failing to make the most of two headed chances in the first 20 minutes.
Burton came to life as Andy Ducros began in midfield. He was denied the opening goal when his clear shot from 25 yards was saved at the second attempt by Ryan Clarke.
Ducros had the hand in the decisive goal on the stroke of half-time and, when he pounced on a loose ball to set up Harrad a cracking finish from the edge of the area.
With Substitutes Guy Madjo and Bruno Teixeira livening up their attack, Forrest Green dominated the second half.
Darren Jones had a stinging 25-yard shot pushed onto a post by Deeney in the 65th minute and his brilliant finger tip save kept out Teixeira's 12-yard shot on 78 minutes.
Three minutes from time Teixeira was guilty of a glaring miss when he headed onto the top of the cross bar from three yards with the goal at his mercy.