Paul Hartley gave the hosts a first-half lead before Ian McCall's side equalised through Jim McIntyre in the second period.
However, United defender Mark Wilson scored a howler of an own goal and Kevin McKenna put the game beyond doubt with a powerful header.
Hearts carried on from where they left off at Easter Road last Sunday and started the game in dynamic fashion. Phil Stamp played the ball to Dennis Wyness, who was lurking in the box, and the striker's deft shot was tipped over the bar by Paul Gallacher after just two minutes.
The game came to life in the 15th minute when United hitman Collin Samuel eluded the challenge of Steven Pressley on the edge of box after a Jim McIntyre flick on, but he was denied by a superb save from Craig Gordon.
And it was the home fans who were celebrating after 17 minutes. Playmaker Stamp threaded a delightful pass through to Hartley in the box, after linking with Wyness, and he showed great composure to slot the ball low into the corner of the net.
Hearts shot-stopper Gordon had to look alert to get down to a fierce Wilson free-kick as the away side looked for an equaliser. Just seconds later, the former Scotland Under-21 keeper saved a powerful shot from the same player.
Whatever United manager Ian McCall said to his players at half-time it seemed to have worked as they levelled in the 53rd minute. McIntyre eased past McKenna in the box and drilled the ball under Gordon into the net.
United were inspired by their equaliser and came close to taking the lead for the first time in the game. Samuel, who scored a hat-trick against Hearts for Falkirk last season, raced clear down the flank but Canadian international McKenna recovered to superbly block his shot.
Hearts restored their lead after Wilson put the ball through his own net after 65 minutes.
De Vries played a relatively harmless ball across the six-yard box but the defender made a real mess of trying to clear the ball and sliced it into the net.
United lost any hope they had of winning the game after Wilson's error and Hearts scored a third in the 73rd minute through McKenna. The defender, who was arguably at fault for the visitor's goal, directed Hartley's brilliant diagonal cross into the far corner of the net past the hapless Gallacher.
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