Thokozani Sekotlong fired the hosts in front after 28 minutes and they looked like they might hold on, until the veteran striker equalised with seven minutes to play.
Nomvethe then lashed in a second two minutes into stoppage time as the Birds climbed out of the relegation zone and into 13th place.
Both sides showed plenty of energy during the early exchanges, but there was a clear lack of quality in the final third.
Swallows had the first chance when Lerato Chabangu laid the ball into the path of Vuyisile Wana in a training ground routine from a free-kick, but the striker blazed over.
United had their first chance midway through the half when Kwanda Mngonyama returned the favour from 20 yards as his strike ended wastefully on the vacant grass beyond the goal.
But the game burst to life in the 26th minute when Terrance Mandaza struck the base of the right upright after spinning and shooting sharply from the centre of the box.
Nhlanhla Vilakazi was the architect of that move and the Team of Choice attacker was again at the heart of play for the opening goal, dispossessing Sandile Sibande in midfield and running in on goal, before laying the ball left to Sekotlong, who arrowed into the far corner.
The Birds’ best chance to score came in stoppage time at the end of the half when Chabangu’s attempted scissor-kick ballooned over after a good cross from Dikgang Mabalane.
Swallows had their best opening 11 minutes into the second half when Rudi Isaacs floated a free-kick into the near post from the right, with Luvhengo Mungomeni heading wide.
The pressure continued to come from the Soweto side, who then had two Chabangu shots struck centimetres wide as they pressed for an equaliser.
There was an even better chance to follow for Fani Madida’s side as Mungomeni floated in a wonderful ball from the right that picked out substitute Siyabonga Nomvethe, who somehow headed wide.
Another sub Mohammad Anas shot across the face of goal at the other end — easing some of the pressure — but it was Mabalane that missed another fine chance, shooting off target after Nomvethe’s flick put him through.
Swallows were beating down the door at that stage and were duly rewarded when an unmarked Nomvethe head in from Mungomeni’s cross.
After Anas squandered another gilt-edge one-on-one chance, Nomvethe silenced the home crowd by latching on to Chabangu’s through ball and completing a low finish.
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