Jackie McNamara's side sealed a fourth win over their Ayrshire opponents this season after coming from behind in the second half to secure full points.
Gary Mackay-Steven gave the hosts an early lead but Alexei Eremenko equalised in the run-up to half-time before Kris Boyd put the resurgent visitors ahead after 63 minutes.
However, Curtis Good headed home a quick-fire equaliser for the Taysiders and John Rankin completed the comeback with a spectacular 71st-minute winner to build on their previous weekend's win over St Mirren in the Scottish Cup.
United created the first goalscoring opportunity after five minutes and Nadir Ciftci was unfortunate to see his effort come back off the woodwork.
He cut in from the left and beat both Jackson Irvine and Lee Ashcroft before firing in a shot that was too good for Craig Samson but came rolling back off the inside of the post.
However, just five minutes later the hosts were ahead.
Irvine slipped as he attempted to pass the ball back to his keeper from Ciftci's pass forward and Mackay-Steven seized the opportunity, taking a touch to settle himself before slotting this measured finish low into Samson's far corner.
Midway through the first half, Ciftci had a half-chance to double his side's lead but he could only shoot straight at Samson.
A few minutes later, Kilmarnock almost equalised.
Irvine's cross was met by Boyd whose shot on the turn from 10 yards was brilliantly saved by United keeper Radoslaw Cierzniak.
From the rebound, Killie's Jeroen Tesselaar was first to react but he saw his effort come crashing back off the post.
The visitors were stepping up the pace and soon after Rory McKenzie's cross found Eremenko but again Cierzniak proved equal to the task by making a diving save.
With 10 minutes to go before half-time, Ciftci's pass picked out the on-rushing Andrew Robertson who drilled in a shot which was not far away from Samson's far corner.
Then, with their next move up the field, Rankin's ball across the face of the goal almost brought United a second goal but Keith Watson was just inches away from connecting as he threw himself full length.
But after 40 minutes Kilmarnock levelled in style.
McKenzie's header down set up Eremenko who got the ball under control before unleashing a 25-yard effort which beat Cierzniak low to his left.
Early in the second half, United made a double substitution as Ryan Dow came on for captain Stuart Armstrong while Farid El Alagui replaced Ryan Gauld, but it was Killie who came closest to the next goal on the hour mark.
Irvine's deep cross found Boyd and just as he was preparing to pull the trigger, United defender Watson got across to block his effort.
At the other end, Robertson surged up the park for the hosts and saw his left-foot drive sail over Samson's bar.
But Killie got their noses in front after 63 minutes.
Craig Slater's ball forward was met by Boyd who lobbed the advancing Cierzniak from the edge of the penalty area as his effort just crossed the line for his 16th goal of the season.
Their lead was short-lived, though.
Two minutes later, on-loan Newcastle defender Good glanced a header from a Mackay-Steven corner into Samson's far corner to make it all square again.
And United regained the initiative 19 minutes from time - and this time it was for good.
Ciftci released Rankin and from an acute angle he lashed it past Samson who was helpless.
United held out despite Killie's best efforts to equalise towards the end for an important three points.