A stunning strike by Keith Watson in the fifth minute put United ahead, but the visitors suffered a blow when Nadir Ciftci was shown a straight red card following an incident involving Gary Mackay-Steven which saw three other players booked.
Gary Warren headed home a 54th-minute equaliser before Draper dramatically won it right at the end for the home side.
Inverness manager Terry Butcher named the same starting line-up that began the ill-fated Highland derby on Friday which had to be abandoned due to floodlight failure.
United showed three changes from the team that beat St Mirren 4-0 in the Scottish Premiership on Saturday.
John Souttar and Mark Wilson were not involved, while Brian Graham dropped to the bench. Watson, Sean Dillon and Stuart Armstrong all came back in.
It was no surprise after such a lively start that the opening goal took just five minutes to arrive.
Ryan Gauld's run across the 18-yard box eventually saw him set up Watson, who from an acute angle crashed an unstoppable shot into the top corner, much to the delight of what was a sizeable away support.
The home side's response to falling behind was a positive one with Billy McKay inches away from turning home Doran's low ball across goal after he had been played in on the right by Shinnie.
Doran was involved again moments later, this time striking at goal from 25 yards having been teed up by Marley Watkins, but it was held by Cierzniak.
Watkins was involved again soon after, his cross being played back to Doran by McKay - only for the Irish winger's shot to fly a yard wide.
It was all Inverness and only a point-blank save by Cierzniak kept out Watkins' back-post header from Tremarco's excellent cross.
Inverness continued to press and Cierzniak had to be lively again as the interval approached, turning an effort from 18 yards by Nick Ross round the post after good work by Draper.
The biggest talking point of the opening 45 minutes, however, arrived a few minutes before the break following an incident involving Richie Foran and Mackay-Steven, who went down in front of the home dugout.
In the ensuing melee, United's Ciftci was shown a straight red card, while team-mates Gavin Gunning and Paul Paton were cautioned, as was the home side's Tremarco.
Referee Kevin Clancy was not finished there, though, and Butcher was sent to the stands along with United first-team coach Darren Jackson.
United brought on Graham and Morgaro Gomis at the interval, but the 10 men were soon finding themselves on the back foot, Cierzniak doing well to push away Shinnie's teasing cross as McKay threatened to turn it in.
It was only delaying what you felt was the inevitable and in the 54th minute Inverness drew level.
John Rankin's foul on Watkins gave Shinnie a chance to whip a free-kick in with his left foot that Warren powered home from close range.
United, however, almost regained the advantage with a shot from 18 yards by Paton that looked to be heading into the top corner until Tremarco diverted it away for a corner via his head.
At the other end, Doran sent in a wonderful ball from the left, but McKay could not get the decisive touch under pressure from Cierzniak.
Inverness had strong claims for a penalty in injury-time when Foran looked to be held down by Sean Dillon, but the match official was not interested, and proceedings subsequently entered extra-time.
McKay had the best chance of the first 15 minutes, sending a header over after getting on the end of Tremarco's cross.
That was the only real opportunity of note until the 120th minute when substitute Danny Williams' cross was headed in off the underside of the bar from close range by Draper.