The midfielder netted a drilled effort following Francisco Sandaza's cutback in the fourth minute to set the tone for the SPL clash.
United totally dominated the first half and should easily have been further ahead at the break, particularly through good chances which fell to Sandaza and Andis Shala.
Although Hamilton rallied somewhat in the second half, they were unable to salvage a point.
Hamilton welcomed back goalkeeper Tomas Cerny as they made five changes.
Cerny and midfielder Simon Mensing returned from suspension and were joined in the starting line-up by Trent McClenahan, back from injury, veteran Chris Swailes and Paul McGowan.
The three changes made by United manager Craig Levein included one he was forced to make due to the loss of Jon Daly, who suffered cruciate ligament damage early in the previous weekend's draw against Hibs.
Shala made his first start for the Tannadice side while Paul Caddis and David Robertson were promoted from the substitutes' ranks.
United looked positive from the outset and took the lead after just four minutes - Sandaza was alert in chasing a ball heading for the byline and kept it in with a cutback to Craig Conway, who drilled a powerful diagonal drive low into the left-hand corner of Cerny's net.
The visitors had an excellent opportunity to make it 2-0 after 18 minutes when Prince Buaben set up a dangerous attack with a ball over the top into the penalty box.
Sandaza had his shot blocked, Shala arrived at pace to try his luck with the rebound, which was stopped by the stretching Cerny, but still Hamilton failed to clear their lines and the chance was only finally lost when Robertson booted a third attempt well off target.
Paul Dixon's drilled free-kick was deflected past the right-hand post and Conway forced Cerny into an impressive reaction save as he took Dixon's pass in his stride and shot as he romped down the left flank, while shortly before the half-hour, Wilkie lifted an effort over the bar from almost on the goalline.
As Sandaza waited on the sidelines to return after treatment following a James McArthur challenge just before the break, Shala headed Dixon's cross from the left onto the bar.
Hamilton introduced Joel Thomas and James Gibson in place of McGowan and Easton at the break, reshaping to a more attacking 3-5-2.
They were immediately much improved and excellent link play between McCarthy and Mensing on the edge of the area allowed the latter to try a diagonal effort which brought the first real save of the game from Zaluska.
United took 12 minutes to record their first chance of the second half when Conway lined up over a 25-yard free-kick, weaved his way closer to goal and shot, but his effort went well over the bar.
Sandaza rolled the ball cross the edge of the box to Robertson, but his shot was safely claimed by Cerny; the Spaniard then played in Shala moments later but again the Hamilton goalkeeper was not undly troubled in diving to smother the shot from the left.
Cerny prevented the waiting Shala and Robertson converting a useful Dixon cross as the game entered its final quarter, while Hamilton came close to an equaliser moments later when Thomas stabbed at McClenahan's ball from the right byline and sent it wide of the opposite post.
Although the second half lacked the number of goal chances seen in the busy first, Accies battled for an equaliser in the closing stages and were almost gifted an opportunity when Derek Asamoah's cross stuck between the feet of Garry Kenneth, but Thomas found himself unable to pounce on the loose ball.
McArthur had a drilled effort saved from McCarthy's cross and Asamoah weaved into a promising position before firing wide of the left-hand post.
But United hung on and clinched the three points which they had earned in the first half when Zaluska made a crucial close-range block from McCarthy deep into stoppage-time.