Assured of a place in Europe before a ball had even been kicked, following Aberdeen's defeat at Rangers, buoyant Hearts surged at their opponents from the off.
Lee Wallace grabbed an impressive opener after 11 minutes - only his second goal of the season - to end the club's four-game drought.
Bruno Aguiar then netted after 26 minutes to effectively end the contest, before Eggert Jonsson's late penalty sealed only Hearts' second win in nine matches.
Former Hearts manager Craig Levein has enjoyed happier afternoons in Edinburgh and his fourth-placed Taysiders will still require a point next weekend to guarantee their own European spot.
Marius Zaliukas dropped out through suspension for Csaba Laszlo's side, along with injured Laryea Kingston.
Christos Karipidis and Ruben Palazuelos were restored while Andrew Driver remained absent.
United pair Sean Dillon and David Goodwillie came in for Mihael Kovacevic and Danny Swanson.
Given Hearts' European qualification, there was a feel-good factor around the stadium and it was almost helped by an early goal from the hosts.
Wallace's fine run and cross down the left picked out Aguiar whose sharp eight-yard shot was straight at goalkeeper Lukasz Zaluska.
Laszlo's team appeared in the mood and continued their lively opening when Lee Wilkie blocked a Calum Elliot effort, before David Obua saw a shot deflected wide from Palazuelos' centre.
But a deserved breakthrough was only delayed until the 11th minute. Elliot held the ball up superbly in the penalty area and the on-rushing Wallace burst through to collect and strike a sweet finish across Zaluska.
United, backed by a strong away support, replied when Marian Kello turned Goodwillie's strike around the post, before Prince Buaben flashed a shot wide from distance.
But Hearts were soon back on the front foot and Aguiar curled a dipping free-kick over the bar.
United claimed for a penalty on 25 minutes after Goodwillie went down just inside the area under pressure from Jonsson, but referee Brian Winter was unmoved and instead spoke to the young striker.
Within seconds, Laszlo's men had rushed up the park to net their second. Elliot did well on the right to cut back for Aguiar who clipped the ball beyond Zaluska at the near post for his seventh of the season.
David Robertson fired over as United sought a foothold in the game, before another failed penalty claim on 35 minutes again riled the Tangerines.
Francisco Sandaza went down trying to turn Karipidis, but Winter instead cautioned the Spaniard for diving.
Levein's charges were in need of an early reply after the interval, but Hearts appeared content to stay compact and frustrate them.
The first opportunity of the half fell to Elliot whose close-range overhead kick was parried by Zaluska.
Craig Conway's 55th-minute free-kick was deflected wide by Jonsson, but United were generally struggling for openings and Palazuelos was next to threaten for Hearts from long range.
Levein made a double change to spark more urgency from his side just after the hour mark. Andis Shala came on for Sandaza and Swanson for Robertson.
But the match became a scrappy affair as Hearts held on to their advantage, with Garry Kenneth picking up a booking for his obvious displeasure at proceedings.
Elliot had a glorious chance to increase the home side's lead on 70 minutes but his volley soared over.
As United pushed more players forward, Kello did well to turn away a dangerous Kenneth cross before Morgaro Gomis struck a 25-yard drive straight at the goalkeeper.
However, it was game over for the visitors 11 minutes from time.
Elliot was pushed over by Kenneth in the box and Jonsson stroked home the spot-kick. Home substitute Christian Nade could have even added further gloss to the scoreline late on, but blasted high and wide.