Dods rose to meet it but Aberdeen goalkeeper Jim Leighton palmed away his header.
Aberdeen were set to take a draw after Darren Dods had levelled Robbie Winters' opener.
Dods had originally been a target for Levein's predecessor at Tannadice, Craig Brewster.
BBC: SPORT | Football | My Club | Dundee Utd | Levein delight at capture of Dods
Moments later, Cerny blocked Dods' shot after Lee Wilkie and Scott Robertson combined to open up the Hamilton defence.
Dods, previously with Hibernian and St Johnstone, joined Inverness in July 2004 and currently captains United's Scottish Premier League rivals.
BBC: SPORT | Football | My Club | Dundee Utd | Levein delight at capture of Dods
United, looking to build on last weekend's 4-1 thumping away to Falkirk, brought Darren Dods into defence for the injured Andy Webster.
Darren Dods and Danny Grainger took the places of Lee Wilkie and Paul Dixon after United's exhausting semi-final defeat by Celtic on penalties on Wednesday.
Darren Dods conceded a free-kick on the edge of the United box and, though Papac's initial effort was cleared, Mendes lashed the ball home with a stunning diagonal drive.
United were profiting from cross balls, and Darren Dods perhaps should have done better when Chris Smith flapped at a Robson cross, but the defender put his header well over.
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.
The game could have been all but over a few minutes after the restart when Swanson made an unchallenged run before laying off for Conway, but his shot was inadvertently blocked by team-mate Darren Dods.
If they were to smash that dismal record, the Perth Saints would have to get past United defenders Scott Severin, Garry Kenneth and Darren Dods, the latter a member of that Saints team a decade ago.
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