Playing with a strong wind behind them, Hungerford had the better chances in a well-contested first half.
Jamie Gosling shot straight at James Nortei from 20 yards and Raymond Spence got around the back of the Cleeve defence but failed to get the ball past the advancing Nortei.
In reply Jake Lee
had clear header from an Alex Sykes' free-kick but he failed to get it on target.
The Villagers made a strong start to the second half and Jake Lee
headed down a Sam Avery cross to Mike Tambling but he couldn't keep his volley down.
The winner came in the 67th minute after Sam Avery cleared to Carl Brown.
Brown was shadowed by three defenders wide outside the Hungerford penalty area but turned inside to make space for himself and unleashed a right-foot shot from 20 yards which beat Daniel Lincoln.
Hungerford had good chances to equalise, Dean Stowe testing James Nortei with a free-kick from 25 yards and Harry Goodger getting a strong header to a Stowe cross only to be denied by the finger tips of the diving Nortei.
With Hungerford pushing forward, the Villagers should have finished the game off when they caught them on the break twice, but the final passes failed to make them count.
Alex Sykes' men now continue their testing month when they travel to fifth-placed Paulton Rovers this Saturday and entertain league leaders Bideford the following week.
Cleeve: J Nortei, J Crowford, R Clarke, L Smith, S Avery, M Pritchett, M Tambling (E Kennedy, 81), M Jackson, J Lee (S Midwinter, 89), C Brown, A Sykes. Subs not used: A Mace, S Jones.