The two sides couldn’t be separated during the 90 minutes in the Highlands with the match ending 0-0, though the home side had the better of the chances throughout.
Raith will now meet Kilmarnock in the last four after the Ayrshire side beat Hamilton 3-2.
Raith had the better of the opening 10 minutes without troubling Caley keeper Cameron Mackay, before the home side came back into the game and almost took the lead on 14 minutes when midfielder Reece McAlear fired a shot just narrowly wide of Thomson’s goal.
Six minutes later and Inverness went close again when striker Lewis Jamieson shot low from the edge of the box but Thomson got down well to save.
On 24 minutes Raith had a let off when captain Kyle Benedictus headed the ball into his own net but his blushes were spared as the linesman flagged for an offside.
There was a blow for the visitors on 33 minutes when the lively Ethan Ross went off injured to be replaced by Blaise Riley-Snow and five minutes later Aidan Connolly should have done better when he was played through by Poplatnik but his final effort was weak and Mackay saved comfortably.
The same player delivered a good free kick two minutes later which was met by Benedictus but Mackay saved the defender’s header.
Early in the second half Raith made a double change with Poplatnik replaced by Ethon Varain and Kieran Mitchell coming on for Dario Zanatta.
The game fell flat with neither side able to create anything meaningful in an attacking sense until the 68th minute when the home side passed up a glorious opportunity to open the scoring.
Caley broke two on one and Tom Walsh played in Jamieson but his attempt to lift the ball over Thomson was brilliantly saved by the Raith goalkeeper.
With three minutes to go Connolly went close with a curling shot from 20 yards which went narrowly wide.
It was the last attempt from either side and with the match ending in a stalemate it went to penalties.
With both sides having missed one each the shootout went to sudden death and, after Liam Dick scored for Raith, Thomson saved Shane Sutherland’s effort to send the Kirkcaldy side through to the last four.