Boro came into Tuesday night's League One match at Broadhall Way in dismal form, but earned a deserved victory when Haber stole in behind the Bees' defence following a long header upfield from skipper Mark Roberts.
The hosts started the brighter and almost went ahead in the fifth minute when former Brentford midfielder Gavin Mahon headed against the post.
It was then Haber's turn to strike the woodwork, heading Greg Tansey's corner against the crossbar, while he was again denied after 35 minutes when visiting goalkeeper Simon Moore kept out his close-range effort.
Uwe Rosler's side came flying out of the blocks after the break and almost levelled when Clayton Donaldson found himself one-on-one before Haber's opportunist goal broke the deadlock.
Bradley Wright-Phillips saw a good chance to level saved before Sam Hoskins warmed Moore's palms late on at the other end as Stevenage held on for the win.
Brentford missed out on the chance to move into the top three after suffering a first league defeat in six matches.