The goals came 86 minutes apart, with an early strike from Ross remaining the decisive moment until Doran's last-minute effort set up a home quarter-final against Dundee United.
Ross opened the scoring in this fifth round replay thanks to a touch of good fortune when Scott Robertson's attempted clearance for Stranrarer struck the Inverness midfielder with the ball which rebounded beyond onrushing keeper David Mitchell.
And, as the visitors chased an equaliser, the hosts found space to seal the tie with Doran firing into the bottom right-hand corner from outside the box after collecting James Vincent's assist.
However, the cup kings did not have it all their own way at the Caledonian Stadium with Steven Bell's late long range effort forcing a great save out of Dean Brill when the margin was just a single goal.
Twice Grant Gallagher tested Brill from distance during the second half, while Inverness should have added further goals before the break when Greg Tansey and Graeme Shinnie went close - the former producing a great save from David Mitchell.
The match had come close to being called off, with the sodden Highlands pitch needing two inspections before the referee Alan Muir declared the surface playable.
In the end Inverness did enough to continue their excellent form in the knock-out events this season, having already assured themselves of a place in next month's Scottish League Cup final against Aberdeen.