County keeper Michael McGovern then produced a brilliant reaction save to deny striker Jon Daly.
But United were a different team after the break and Jon Daly's header and Danny Swanson's volley restored parity.
United's best chance fell to Jon Daly but the striker could not hit the target after finding space in the box.
The visitors soon found their feet and Jon Daly tested Darren Randolph with a low shot but the keeper was equal to it.
United started the quarter-final replay on the front foot and Jon Daly stung the palms of goalkeeper Allan McGregor with a fierce early drive.
Dundee United looked to put the game to bed and David Robertson volleyed over after Jon Daly had caused havoc in the Inverness box.
Jon Daly headed United into the lead from close range when Paul Dixon's shot was initially blocked by goalkeeper Tomas Cerny after 20 minutes.
Striker Jon Daly is currently joint top of the League Two scoring charts, thanks in large part to a burst of seven goals in eight days.
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Hamilton took the lead on Tayside through Flavio Paixao's header, but David Goodwillie equalised before the break and Jon Daly struck in injury time to give United victory.
United almost took the lead with their first meaningful attack in the 21st minute, when Jon Daly's bullet header was cleared on the line by the alert Rankin.
Conway's free-kick found the unmarked Jon Daly in the fourth minute, but the striker could not get the ball under control as it trickled back to Hearts goalkeeper Jamie MacDonald.
Barry Douglas made a fine United debut following his summer switch from Queen's Park, while Danny Swanson, Jon Daly and David Goodwillie were also recalled after Thursday's Europa League exit.
That incident marked Aberdeen's seventh sending-off already this season, but it had the effect of jolting the game back to life and Langfield was tested by both Craig Conway and Jon Daly.
United swapped midfielder for striker, with Morgaro Gomis coming in for the suspended Jon Daly, while it was the opposite for Celtic, with Stokes replacing the rested Ki Sung-Yeung in the starting line-up.
But Hearts saw out the remainder of the game - including five minutes of injury time - with Balogh making a comfortable save from Dods after substitute Jon Daly had laid off Caddis' clever pass.
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