The Tangerines, looking to return to the top flight after a one-season absence, were themselves a player light at Tannadice Park due to a red card but in the end neither side could do enough to find a way through.
Dundee would have been a goal down soon after if not for a fine Cammy Bell save, getting his trailing foot to deny Bingham's shot from a one-on-one position.
It was Remi Matthews's turn to produce a big stop a quarter of the way through under pressure from Thomas Mikkelson, with neither team able to show the killer instinct in front of goal.
Hamilton grew into the match more and more as it went on, and they came close to opening the scoring when Templeton worked some space and sent a deflected shot inches wide of the far post.
Matthew again did well to turn a Mikkelson header wide as United - now unbeaten in their last 11 games - looked to turn the screw, while Mark Durnan blasted over the crossbar on the hour.
The Accies looked to have squandered their big chance to take a lead into Sunday's second leg after Dougie Imrie failed to convert a cross into the box, although a second yellow shown to Simon Murray for supposed simulation with more than 10 minutes left to play gave them further hope.
Incredibly, though, injuries to Imrie and Massimo Donati saw the visitors down to nine men for the closing stages, and with neither side able to break the deadlock it remains all to play for at New Douglas Park.