It could have been a very different story had Inverness capitalised on an early opportunity when Liam Polworth released Miles Storey, but his low effort was kept out by the leg of County goalkeeper Gary Woods.
The visitors improved as the first half progressed and almost broke the deadlock in spectacular fashion when Martin Woods rattled the crossbar from 30 yards out.
Poor marking gifted County their opening goal as Jackson Irvine took advantage of space in the box to power a header home from a corner.
Fon Williams was then forced into action with a fingertip save to keep out a powerful strike from Craig Curran.
After remaining on the back foot for long spells, Inverness finally threatened to get back in the game when Ryan Christie's clever flick came back off the inside of the post.
A lifeline remained with just over 10 minutes left in the match as Greg Tansey unleashed a swerving strike which somehow drifted past Woods as the hosts set up a dramatic finish.
However, the County defence reacted well to keep it tight in the closing stages as the travelling supporters celebrated a semi-final spot being secured.