Experienced playmaker Kurt Lentjies got the goal in the ninth minute, bravely stabbing home from close range following a free-kick to the back post by Deolin Mekoa.
Victory saw United leapfrog Polokwane City in the standings, while Celtic - who thrashed champions Mamelodi Sundowns 5-0 in midweek - remain in fifth spot with four games left.
The top-eight finishers in the Absa Premiership qualify for next term's MTN8 Cup, which was won by log leaders Kaizer Chiefs this season.
Lentjies's early goal prompted the hosts to push forward, with Keagan Ritchie heading straight at Virgil Vries just past the half-hour mark off a delivery from Musa Nyatama.
Phunya Sele Sele midfielder Keagan Buchanan then shot over on the turn moments before Golden Boot hopeful Lerato Lamola poked an effort wide just before the break.
The second stanza followed a similar script, with Lamola nearly drawing level at the death but his blocked strike was gratefully gathered by Vries.