Manager Neil left Hamilton to take over at English Championship side Norwich City on Friday, and his absence was felt in an exciting encounter at New Douglas Park on Monday.
The home fans displayed their gratitude to Neil with a minute's applause early in the first half.
However, it was United that took command of the Scottish Premiership clash with two goals either side of half-time.
Midfielder Stuart Armstrong gave the visitors the lead with a powerful strike from 22 yards out in the 38th minute, and then turned creator in the 46th minute as he played Gary Mackay-Steven through to double the advantage.
Defender Jesus Garcia Tena gave Hamilton's caretaker boss Martin Canning hope with a sublime curling free-kick 11 minutes later, and the comeback was completed in the 70th minute by Ali Crawford.
Crawford hooked home on the rebound after initially being denied by Radoslaw Cierzniak following Tony Andreu's defence-splitting pass.
But there was to be one last twist in the tale in the 77th minute.
Centre-back Dillon rose highest to power a header home from a corner, securing all three points and ensuring United leapfrog Hamilton and Inverness Caledonian Thistle into third place.
