The visitors had begun the cup-tie brightly enough, Kris Boyd, released by Craig Slater, fired the visitors' first chance just wide of Radoslaw Cierzniak's near post after 20 minutes.
United, looking for a third successive win, soon took advantage when surging up the park for left-back Robertson to take a touch in controlling Armstrong's pass before beating Samson in the Kilmarnock goal from 10 yards out.
Whatever manager Allan Johnston said to his strugglers 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 Slater's free-kick completely.
Darren Barr could not believe his luck but still he had to show composure as he produced an expertly executed overhead kick to hook the ball into an empty net.
The Rugby Park side went one step further by gaining the ascendancy in the 49th minute through Johnston's sublime 20-yard curler into Cierzniak's top corner.
However, Scotland forward Boyd was still keeping the United defence on their toes before the hosts equalised 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.
With 15 minutes to go Robertson's second made it 3-2 with a low shot from teenage sensation Ryan Gauld's through-pass, which beat Samson at his near post, before substitute Brian Graham's penalty, after Gauld was brought down inside the box by Gabriel Reuben.
A blistering Gary Mackay-Steven strike six minutes from the end completed the rout and safe passage for Jackie McNamara's side.