Goals from Lee Wallace, Bruno Aguiar and Eggert Jonsson, from the penalty spot, proved more than enough against a disappointing United.
Hearts took control from the start and a great run from Wallace set up Aguiar but he directed his shot straight at Lukasz Zaluska and the goalkeeper blocked it.
United struggled to contain the adventurous Wallace in the early stages and the full-back put his side ahead in the 11th minute with a well-worked goal.
He played a one-two with Calum Elliot before bursting into the penalty area and sending a low shot into the far corner of the net.
United tried to hit back but a long-range effort from Prince Buaben was over-ambitious and the ball finished yards wide.
Hearts looked more threatening when they went forward and a 22-yard free-kick from Aguiar dipped just over the crossbar midway through the first half.
It was no surprise when Aguiar fired Hearts two ahead after 26 minutes.
Robbie Neilson released Elliot with a quick throw-in and the striker cut the ball back to the edge of the six yard box. Lee Wilkie was slow to react and Aguiar nipped in ahead of him to clip the ball past Zaluska.
David Robertson almost pulled a goal back for United 12 minutes from the break when he was given too much time to bring the ball down in the area, but he lashed his shot a foot too high.
Moments before half-time, David Goodwillie tried his luck from 30 yards after spotting Marian Kello off his line but again the ball dipped over.
Hearts kept the upper hand at the start of the second half and Elliot and Ruben Palazuelos both had decent efforts rebuffed.
Michael Stewart then tried to chip Zaluska but the goalkeeper was alert to the situation and clutched the ball underneath his crossbar.
Elliot should have made it 3-0 after a delightful through ball from Aguiar but he volleyed hurriedly over when he had time to do better.
But the striker was tripped late in the game by Garry Kenneth and referee Brian Winter awarded the penalty after his linesman flagged.
Jonsson stepped up to send Zaluska the wrong way with the spot-kick and put the game beyond doubt.