Radoslaw Cierzniak made a mess of the Scotland Under-21 striker's long-range shot, which slipped under the United goalkeeper and crossed the line early in the second half at Tannadice.
Cierzniak's mistake proved to be costly as Jackie McNamara's side were unable to respond and dropped below Inverness Caledonian Thistle into fifth place in the SPL table.
McNamara and St Johnstone boss Tommy Wright were sent to the stands following a touchline altercation after United substitute Farid El Alagui went down in the penalty area, but the referee waved play on.
United winger Gary Mackay-Steven was also denied a spot-kick in stoppage time in a frustrating night for the home side.
Victory for the visitors was their first away from home since January in the top flight and ensured that they move four points clear of seventh-placed Hibernian in the battle for a top-six finish.
Hibernian boss Terry Butcher had to settle for a goalless draw at his former club Inverness in the other Premiership game to be played on Wednesday,
Inverness were uninspiring in their final game before Sunday's League Cup final against Aberdeen, but the point moves them up a place in the table and they are now six points adrift of third-placed Motherwell with a game in hand.
'I’d probably pick Cole Palmer alongside Bellingham and Rice in there at the moment. For me, Declan Rice is good enough to play that central midfield role on his own' Thomas Tuchel advised to DROP Manchester City star by England legend
Manchester United staff threatened with sacking for leaked information... in a leaked email
