Visitors United surged ahead as early as the fifth minute at the Tulloch Caledonian Stadium through Keith Watson. The full-back collected a cross-field pass from Ryan Gauld and struck home past Dean Brill.
Midfielder Doran then tried his luck with a long-range effort for Inverness, but Radoslaw Cierzniak made a good save between the visitors' posts.
United went down to 10 men three minutes before the half-time interval as Nadir Ciftci was sent off following a melee between the two sets of players after Gary Mackay-Steven had gone down under a challenge from Inverness captain Richie Foran.
Inverness created chances to win the tie during the remainder of the second half, but failed to prevent the contest from going into extra time.
The hosts eventually booked their place in the last four of the competition in the final seconds of the 30-minute period. Danny Williams sent in a cross at the back post that Ross Draper headed in off the underside of the crossbar.