Goals from Liam Polworth, Riccardo Calder and Daniel Mackay put the Highlanders into the last-eight draw, to be made on Tuesday.
It was their first home win since mid-July and Robertson said: “It is a good day’s work. We have been working very hard on our defensive aspects and that is our second clean sheet.”
Inverness got off to a flying start as midfielder Polworth saw his deflected shot from just outside the area beat keeper Greg Fleming after just 90 seconds.
Rory McAllister almost replied against his former employers but his overhead kick was too high to trouble Owain Fon Williams.
They were growing into the game and Jamie Stevenson’s long-ranger after 20 minutes required a decent stop from the Inverness keeper before a deflected Willie Gibson effort dipped over the bar.
Six minutes into the second half it was all but game over as Calder traded passes with Polworth before guiding the ball beyond Fleming.
And 16-year-old Mackay wrapped up the scoring in injury-time with a shot from inside the box.