Stuart Armstrong and Gary Mackay-Steven struck either side of half-time to put the Tangerines 2-0 up but Jesus Garcia Tena and Ali Crawford brought the hosts back level at New Douglas Park.
However, Irish defender Dillon netted a 77th-minute winner to help United leapfrog Academical and Inverness Caledonian Thistle back into the top three.
With the first chance of the match after four minutes, Garcia Tena headed an Crawford corner over the bar from 12 yards for the hosts when he appeared to have the time and space to at least hit the target.
At the other end, United striker Nadir Ciftci hooked a shot from the edge of the box over the bar after being set up by Telfer.
In between that, the home fans applauded for a minute or so, apparently in tribute to former player-manager Alex Neil who moved south to take over at Norwich last week, with defender Martin Canning taking over as Accies’ interim player-manager.
Nothing much happened in the goalmouths until the 33rd minute when a free-flowing move from the visitors ended with Charlie Telfer playing in Aidan Connolly, whose angled drive flashed across the Accies goal.
Moments later Telfer fired wide from the edge of the box but Armstrong showed him how it should be done from around 25 yards out, when, after taking a pass from Connolly, he took a touch and rifled a drive low past home skipper and goalkeeper Michael McGovern for the opener (38).
While the PA announcer was telling the pitifully small crowd of 2,384 that Louis Longridge had replaced Stephen Hendrie for the start of the second half for Accies, the Tannadice side went two up.
It was all too easy for Mackay-Steven when he was put in by Armstrong and he slipped the ball past the onrushing McGovern (46).
Accies were down but not out and showed the spirit that had kept them unbeaten in 10 games at home.
Garcia Tena curled in a wonderful free-kick from 25 yards, after Calum Butcher fouled Tony Andreu, and suddenly the Taysiders did not look too comfortable (57).
And there was more drama when the hosts drew level. Andreu’s perfectly-weighted pass split the United defence and although Radoslaw Cierzniak blocked Crawford’s initial shot, the Hamilton player took advantage of the break of the ball to hook it into the unguarded net (70).
But the game swung again towards Tayside when captain Dillon rose at the back post to head in a corner from Chris Erskine, who had replaced Connolly.
Crawford lofted the ball over Cierzniak moments later, but also the bar, much to the frustration of the home support.
As Hamilton left gaps in their bid to get back into the game, McGovern had to make a double save, first from Armstrong and then Erskine, before tipping a Ciftci drove over the bar.
Canning brought himself on for Garcia Tena with two minutes remaining but to no avail, as United held out to move behind Celtic and leaders Aberdeen.