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Warrenpoint had Liam Bagnall sent-off for a reckless challenge on Jim Ervin on 37 minutes, with Willie Faulkner giving the Sky Blues the lead on 69.
Cathair Friel tapped in for his 15th goal of the season six minutes later.
Stephen McCullough put Carrick ahead with a 24th-minute header, while Daniel Kelly made it 2-0 in the 82nd.
Championship leaders Warrenpoint had former Ballymena striker Matthew Tipton in charge for the first time and were holding their own against their Premiership opponents at Milltown until Bagnall's dismissal by referee Keith Kennedy.
Friel had the best of the first-half chances for United, with Stephen Murray and Conor McMenamin going close for the hosts.
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After the interval, McMenamin fired wide from close range and Leroy Millar was also off target, but Faulkner opened the scoring when he picked up the ball following an error by Aaron Traynor and his deflected shot went through the legs of Johnny Parr.
Friel tucked home the second and Faulkner could have grabbed another but his effort was cleared off the line.
At Taylor's Avenue, Ryan Harpur blazed over from six yards out for the Swifts, before McCullough headed in from close range from Chris Morrow's cross for his second of the season.
Dungannon carved out a series of chances throughout the game but found Carrick goalkeeper Brian Neeson in inspired form.
Andy Mitchell, Kris Lowe, Paul McElroy and Andrew Burns squandered opportunities but defender Kelly finished with aplomb for Carrick to ensure his side's passage.
Glenavon and Coleraine will be the other semi-finalists after their dramatic wins over holders Cliftonville and league leaders Crusaders.
Ballymena United and Carrick Rangers secured League Cup semi-final places with 2-0 wins over Warrenpoint Town and Dungannon Swifts respectively.