Kabba, making his first start since the 2003 play-off final after being sidelined with a series of career-threatening injuries, was recalled by United manager Neil Warnock, who was desperate for goals to halt the slide.
Despite another money-spinning FA Cup run the Bramall Lane side had lost six of their last eight league games.
Kabba's recall paid off in the 14th minute when the ex-Crystal Palace striker swept past his marker to run onto Andy Liddell's pass and fire a low shot beyond Ben Williams and into the far corner.
United had been caught out in the opening minutes, with Crewe cutting through the home back four as Michael Higdon turned quickly to fire wide from ten yards.
Kenny Lunt also had a shot blocked by Paul Thirlwell before Kabba struck.
Struggling Crewe, eight games without a win, should have levelled minutes after Kabba's opener. Mark Rivers picked out the unmarked Higdon but his shot had no punch and Paddy Kenny was quickly down to smother the ball.
Kabba weaved his way into the Crewe box on 30 minutes to crack in a low right-foot shot, which was snapped up at the foot of the post by Williams.
Andy Gray then added a second in the 39th minute, side-footing Liddell's low cross past the helpless Williams from eight yards.
United were three up after 66 minutes as Kabba ran onto Liddell's back heel to fire in from 12 yards, before Gray ended a miserable evening for Dario Gradi with a close-range shot nine minutes later.