Goals from Jon Daly, Paul Dixon and Prince Buaben secured a comfortable win for the Tayside club, with Mickael Antoine-Curier's goal proving to be little more than consolation for Accies.
United had been forced to scrape past Ross County on penalties in the last round but never really looked in danger in the early kick-off at New Douglas Park.
They made just one change for the fifth-round tie. Dixon was drafted into the starting line-up after returning from injury and Barry Douglas dropped to the bench following their win over Hibernian.
Accies made four changes following the midweek defeat to St Johnstone, with Mark McLaughlin back in the team after suffering whiplash and Lee Kilday, Marco Paixao and Jon Routledge also given the nod.
Hamilton had a great chance to open the scoring when Antoine-Curier burst down the left flank before squaring into the path of Dougie Imrie but his first touch was too heavy and Dusan Pernis was able to gather.
At the other end, United had the chance to pile on the pressure with a free-kick 25 yards out and Craig Conway aimed for the far top corner with his curling effort but goalkeeper Tomas Cerny was able to punch to safety.
Conway then attempted to turn provider with a cross from the right with David Goodwillie loitering in front of goal, but Martin Canning was able to intercept and clear the danger before the striker could connect.
The breakthrough came after 20 minutes when Conway's quick free-kick was collected by Dixon, whose shot was blocked by Cerny and fell for Daly to send a firm header into the back of the net.
McLaughlin picked up the first booking when referee Craig Thomson punished his trip on Scott Robertson.
The visitors could have added to their lead when Buaben delivered to the feet of David Goodwillie, who turned on the edge of the area and unleashed a decent shot that Cerny did well to smother.
United found the back of the net again after 39 minutes when Dixon tried his luck with a shot from the edge of the box that deflected off Canning past the grasp of Cerny and into the bottom corner.
Paixao could have thrown Accies a lifeline just before the break but his volley was off-target.
Hamilton hauled themselves back into the tie less than a minute after the restart when Andy Graham's cross from the right found Antoine-Curier at the back post and he bulleted a header past Pernis.
But United were on target again with 58 minutes gone. Routledge fouled Goodwillie and Conway flighted the resultant free-kick into the box for Daly, who nodded across goal for Buaben to stab home from close range.
Hamilton made their first change when Paixao was withdrawn to make way for the introduction of Jim McAlister with just over 20 minutes to go.
Daly could have added to his tally for the day with a right-foot drive from 18 yards, which was parried by Cerny.
With time fast running out, Accies threw Damian Casalinuovo and Aaron Wildig into the action in a last-ditch bid to keep their cup hopes alive.
But they failed to find the goals that would force a replay and United held on to ensure their name would be in the hat for Sunday's draw.