The hosts were forced into a goalkeeping substitution at half-time as Craig Samson made way for Conor Brennan with a broken toe.
But Allan Johnston's side were not knocked out of their stride and two goals in the space of three minutes from Tope Obadeyi and Mark Connolly secured a deserved victory on Friday.
United could have led in the 25th minute when Nadir Ciftci headed wide after Sean Dillon got to the byline to chip to the back post and Samson then blocked impressively from Stuart Armstrong.
Kilmarnock came closest to breaking the first-half deadlock when Alexei Eremenko's lofted throughball caused disarray in the United defence and Josh Magennis saw his lob over on-rushing goalkeeper Radoslaw Cierzniak bounce up and onto the top of the crossbar.
Paul Paton drove wide as United looked to gain the upper hand after half-time, but they fell behind in the 63rd minute.
Eremenko produced a deft touch to leave Jaroslaw Fojut in his wake and feed Obadeyi, who, not to be out-done in terms of cute footwork, nutmegged his marker before clinically chalking up a fourth goal of the season
The superb Eremenko was the provider again in the 65th minute, this time as Connolly outmuscled those around him to nod home from a corner.
Substitute Gary Mackay-Steven and Chris Erskine went close as United pressed for a response, but a two-goal win for Inverness Caledonian Thistle over Ross County on Sunday or victory for Hamilton Academical at champions Celtic on the same day will see Jackie McNamara’s men dislodged from the summit.
