Martin Canning's side had the best of the goalless first half by far and should have had a couple of early goals but the home side came back strongly after the break.
However, in the 77th minute Murray, booked in the first half for diving by referee Steven McLean, was adjudged to have gone to ground too easily after a challenge by Scott McMann, although television seemed to suggest he had been caught by Accies defender Scott McMann and should have had a penalty.
Assistant referee Graham Chambers appeared to help McLean with his decision although United can appeal so Murray should be available for the return game at the SuperSeal stadium on Sunday.
An evening of excitement, tension and frustration was preceded by a minute's silence in tribute to those who died and were injured in the Manchester bomb attack.
Accies started with more intensity and purpose, perhaps to the surprise of the Taysiders.
In the seventh minute visiting winger Dougie Imrie stayed on the ground after a challenge by United stopper Mark Durnan as they competed inside the United box for an Ali Crawford free-kick but referee McLean ignored penalty claims
Two minutes later Accies midfielder Danny Redmond missed Rakish Bingham's cross only a couple of yards out, allowing United goalkeeper Cammy Bell to gather.
In the 12th minute Bingham raced on to a David Templeton pass which left him with only Bell to beat but his drive was blocked by the keeper.
Remi Matthews was tested by Murray down at his right-hand post which signalled the start of some Tayside pressure with the Accies keeper forced into a fine reaction save after a mix-up in the Hamilton defence led to an awkward ricochet.
Just after the half-hour mark, Murray was yellow carded for diving inside the box after a challenge by Massimo Donati, back in as captain following Michael Devlin's knee injury
By the interval Canning's men were back on top but with nothing to show for their superiority.
In the 53rd minute Templeton's deflected shot from 14 yards slipped past the post and after United defender Sean Dillon and team-mate Wato Kuate stopped arguing, the corner came to nothing.
Moments later Kuate was replaced by Charlie Telfer and after displaying his anger, he went straight down the tunnel appearing to sarcastically applaud the home fans.
As United fought back, Matthews made a fine save from Thomas Mikkelson's header which was bound for the top corner before Durnan fired over the bar from the edge of the box.
The home side sensed they could get their noses in front for the second leg but in the 75th minute Ali Crawford missed a good chance at the back post, volleying over from substitute Louis Longridge, on for Templeton just after the hour mark
Moments later, Murray was on his way off after being challenged by McMann, with the United striker furious as he trekked off, and the game ended in some acrimony but with neither club any more sure of where they will be playing next season
Source : PA