Cierzniak let May's speculative shot slip through his hands and allowed Saints to consolidate sixth place in the Scottish Premiership as United's five-match winning run came to an end.
Managers Jackie McNamara and Tommy Wright were both sent to the stand as United complained over several decisions from referee Crawford Allan as they searched in vain for an equaliser, notably appeals for a penalty after substitute Farid El Alagui went down.
The goal had come within three minutes of the restart and it was a complete gift.
Firstly, Gavin Gunning's slack pass went straight to David Wotherspoon, who fed May. The Scotland Under-21 striker hit a speculative effort from 30 yards which appeared to be heading harmlessly into Cierzniak's hands.
But the Pole took his eye off the ball and it squirmed out of his grasp and over the line.
St Johnstone took the game to United from kick-off and threatened four times in the opening 10 minutes.
Gary McDonald headed just wide from Wotherspoon's near-post corner and May had a shot pushed wide by Cierzniak after a long ball had caught out Gunning.
Michael O'Halloran then ran unchallenged into the United box but Keith Watson cleared his cutback in the six-yard box, and May volleyed over from 14 yards following a set-piece.
Saints suffered a blow after 20 minutes when left-back Brian Easton pulled up injured. Recent signing Tim Clancy came on for his debut at centre-back and Dave Mackay moved out to the left.
United soon created their first chance when John Rankin's pass sent Gary Mackay-Steven through the middle. The winger took the ball round Alan Mannus but Mackay got back to block his shot with the help of the near post.
They were back on the attack three minutes later when Nadir Ciftci sidestepped Chris Millar after a long ball to give himself a clear shooting chance from 18 yards, but he shot weakly at Mannus.
The rest of the half was a scrappy affair with both sides resorting to long balls more often than their managers would have wanted.
May stunned Tannadice just after the restart with one of the softest goals of the season and United tried to respond immediately, but Mannus saved Ciftci's header and pushed away Ryan Dow's strike.
And Saints were soon back on the front foot when May looped a header on to the roof of the net following a corner.
United brought on El Alagui and Ryan Gauld, who was making his comeback from a hip problem, 20 minutes into the half but Saints came close again through James Dunne's volley.
El Alagui's deft touch set up Mackay-Steven but Mannus was right behind his drive and the substitute was soon involved in the major talking point of the half when he appeared be bundled off the ball by O'Halloran as the forward tracked back.
Ciftci was booked for furiously protesting and referee Allan was soon under fire from the hosts when he did not call play back for a foul on Gauld after the midfielder had shot just wide following a run to the edge of the area.
The incident sparked the touchline row that saw both managers sent off in the 77th minute and the action continued on the pitch as Mannus made a point-blank save from El Alagui following Andy Robertson's low cross.
Saints survived another penalty claim when Wotherspoon challenged Mackay-Steven in stoppage time.
- Sports & Recreation
- Farid El Alagui
- David Wotherspoon
- Alan Mannus
- Dave Mackay