However, Stuart Armstrong drew the sides level again before Robertson grabbed his second of the night to make it 3-2.
Substitute Brian Graham made sure of United's passage into the next round with a well-taken penalty in the 82nd minute after Ryan Gauld had been brought down inside the box by Gabriel Reuben, and Gary Mackay-Steven wrapped up the scoring with a blistering 22-yard drive.
Kilmarnock began brightly enough and after 20 minutes Kris Boyd was released by Craig Slater for the visitors' first chance but he fired just wide of Radoslaw Cierzniak's near post.
They lived to regret not taking that opportunity as United, looking for a third successive win, surged up the park a few minutes later to take the lead in style.
Robertson took a touch to control Armstrong's pass before beating Craig Samson in the Kilmarnock goal from 10 yards out.
Armstrong had another chance with almost the last kick of the first half as the hosts attempted to give themselves some breathing space.
On that occasion, Mackay-Steven went on one of his trademark runs by bursting into the heart of the Killie penalty area from wide on the right.
After leaving two opposing defenders in his wake, Mackay-Steven laid the ball back for Armstrong but rather uncharacteristically he failed to connect properly and the chance went begging.
Whatever Kilmarnock manager Allan Johnston said to his players at half-time had the desired effect as his side powered ahead within four minutes of the restart.
Firstly, in the 47th minute, they had Cierzniak to thank for coming off his line and missing completely Craig Slater's free-kick.
Darren Barr could not believe his luck but still he had to show composure as he produced an overhead kick from six yards at the far post to give his side an equaliser which had scarcely looked on the cards going into the interval.
As if that was not enough, the Rugby Park side went one step further by gaining the ascendancy in the 49th minute.
There was no slice of good fortune this time as Johnston picked possession up from deep and had only one thing on his mind as he surged towards goal.
Johnston made the most of the home team backing off before curling a delightful 20-yard effort high into Cierzniak's top corner.
United had a chance shortly after to get back on level terms only for John Rankin's shot to be saved by the diving Samson.
Boyd was still posing a threat for Killie by keeping the United defence on their toes but it was the hosts who levelled in the 64th minute.
Nadir Ciftci forced his way into the box and looked up to cross for the on-rushing Armstrong who met it perfectly by applying the finishing touch from close range.
Then, with 15 minutes to go, Robertson made it 3-2 with a low shot from Gauld's through-pass which beat Samson at his near post before Graham and Mackay-Steven completed the rout.
- Sports & Recreation
- Darren Barr
- Stuart Armstrong
- Craig Samson