Paterson, 17, Hearts into a 28th-minute lead with a powerful strike from 20 yards and showed a goalscorer's instinct to net the third goal after a casual passback by Sean Dillon.
Arvydas Novikovas had struck three minutes after Paterson's opener with a swerving long-range effort as Hearts bounced back from consecutive defeats by Dundee and St Mirren.
Paterson started the season at right-back before moving into midfield and then the lone striker's role at St Mirren Park last weekend, and he led the line with a maturity beyond his years.
United, who are now without a victory in four games and goalless in three, again missed the creativity of Gary Mackay-Steven and did not threaten a comeback after a brief flurry either side of half-time.
Hearts overcame the pre-match loss of Andy Webster, who failed to recover from a lower back injury, with Darren Barr dropping back into central defence.
Scott Robinson excelled after coming into midfield and there was another youthful element to John McGlynn's selection with 18-year-old Dale Carrick preferred to Andrew Driver on the right flank.
The first moment of note, 15 minutes in, typified a poor start to the game with Hearts goalkeeper Jamie MacDonald spilling Johnny Russell's mis-hit shot behind for a corner.
United were showing some sign of getting an edge with Stuart Armstrong making some decent runs from midfield, but Hearts came into the game as an attacking force midway through the half with Robinson at the heart of it.
The midfielder got the visitors' first effort on target after collecting a headed lay-off by Paterson, turning Willo Flood and curling a bouncing effort that Radoslav Cierzniak pushed wide.
Ryan McGowan headed over from the resulting corner and the right-back was the catalyst for the opener with a bursting run.
Crosses from McGowan and then Robinson were both half-cleared and Dillon could then only block Danny Grainger's long-range strike to the edge of the box.
The ball fell for Paterson, who showed impressive technique to meet it on the bounce with his left foot and drill the ball past Cierzniak.
While United had at least made several attempted clearances before the opener, there was little resistance as Novikovas doubled the lead.
Novikovas was allowed to run at the United defence unchallenged and hit a shot with the outside of his left foot from 25 yards.
Cierzniak stuck out his hand but could not stop the ball swerving inside the post and might have been expected to do more.
Hearts continued to push forward but United regained their composure just before the break and MacDonald made flying saves from Gavin Gunning's header and Armstrong's curler.
United came closer just after the break when Gunning crashed a header off the bar from Ryan Dow's corner before Michael Gardyne fired the rebound just wide from 15 yards.
But the comeback attempts were not sustained, even after captain Jon Daly returned from injury as a 58th-minute substitute, along with teenage winger Ryan Gauld.
The lift of Daly's introduction was quickly nullified by another goal for Paterson just after the hour mark, with the youngster receiving a gift.
Dillon beat Paterson to McGowan's ball down the right wing but passed the ball inside to Cierzniak without looking and put his goalkeeper in trouble.
McGowan challenged the Pole and the ball broke for Paterson to slot in off the far post.
John Sutton should have added to the lead but he steered fellow substitute Driver's cutback wide from six yards, and Cierzniak prevented a fourth when he acrobatically tipped over a Marius Zaliukas volley.