The hosts haven't won an SPL encounter now since dumping Falkirk on February 21, while United's win - their first in three league attempts against Hamilton - takes them a major step forward in their quest for a Europa League slot.
To say it was deserved was something of an understatement. United could easily have had this game in the bag by half-time, such was their dominance, and only Hamilton's excellent keeper, Tomas Cerny, kept them at bay.
United opened the scoring in the fourth minute when Franco Sandaza darted down the right. Believing the ball to have crossed the by-line, Hamilton's defence stopped, allowing Craig Conway the time and space to drill low past Cerny.
Cerny came to Hamilton's rescue in the 18th minute when he blocked first from Sandaza and then Andis Shala.
The Czech keeper then kept shots from Shala and Prince Buaben out in quick succession after 22 minutes.
And he punched clear four minutes later when Conway took a Paul Dixon throw-in on his chest before firing a fierce volley towards goal.
Hamilton came back into the game just before the interval but, although James McArthur and James McCarthy linked up well down the right, Simon Mensing was unable to finish at the near post.
Three minutes before half-time United were denied again when Dixon's cross from the left by-line found Shala, but his header crashed off the crossbar.
Five minutes into the second half Alex Neil and McArthur linked well before McCarthy played a neat one-two with Mensing, but his shot was punched clear by Lukasz Zaluska.
Accies passed up another chance in the 72nd minute when Trent McClenahan raced down the flank and cut the ball back to Joel Thomas, but the striker's shot was well held by Zaluska.
A good United move down the left ended with James Wesolowski seeing a disappointing shot saved three minutes later.
McCarthy played a delightful cross-field pass to McArthur on the right with eight minutes left, but his low effort was saved by Zaluska.
A minute later Thomas sent a shot-on-the-turn towards the right post from the by-line, but Zaluska was alert and punched clear.
Hamilton could have snatched a point five minutes into stoppage time when McCarthy was sent through the middle, but Zaluska raced off his line to block superbly.