Nadir Ciftci gave United an early lead with Stuart Armstrong and Aidan Connolly also on target for the home side, while Manual Pascali netted for Kilmarnock.
United took just six minutes to open the scoring when Armstrong slipped a pass to Ciftci and the striker sent his shot across Craig Samson into the corner.
Kilmarnock levelled the match on 38 minutes when Rory McKenzie’s shot came back off the post and Pascali netted with the rebound.
But United restored their lead just four minutes later as Armstrong burst into the penalty area and beat Samson.
Substitute Connolly secured the victory when he got on the end of a superb long pass from Armstrong and finished from eight yards.
After making a bright start Gary Mackay-Steven almost doubled United’s advantage with a powerful free-kick which forced Samson into a good save.
And United keeper Rado Cierzniak maintained his team’s lead 90 seconds after Armstrong’s goal when he got a hand to a swerving 25-yard drive from Robbie Muirhead which was dipping under the bar.
Samson blocked a Ciftci flick with his feet and Charlie Telfer came close with a shot in the opening minutes of the second half.
Although United continued to dominate the game they were wasteful around the Kilmarnock penalty area until Connolly settled the match in the dying minutes.