In a match where Rudi Skacel was supposed to be the central protagonist on his Tynecastle return, Stevenson put Hearts in front after 15 minutes and added a second from the penalty spot after 31 minutes.
Keith Watson pulled a goal back after 57 minutes but United could not find a second as Hearts climbed above their opponents with victory in their final home match of a tumultuous calendar year.
Stevenson opened the scoring, benefiting from John Sutton occupying the attentions of two defenders and Brian McLean's failure to clear a ball from deep. The ball broke for Stevenson, who simply sidefooted in.
Jamie MacDonald made an instinctive save when Jon Daly met Barry Douglas' free-kick from the left and McLean's torrid afternoon continued as he could only hit the post with the follow up from two yards out.
Dylan McGowan won a penalty when he was bundled over by Willo Flood trying to meet brother Ryan's long throw. Stevenson took the resulting spot-kick and sent Radoslaw Cierzniak the wrong way to score.
But United got back in it when a Barry Douglas free-kick was only half-cleared, Rudi Skacel sliced a shot into the ground and Watson turned to fire into the top corner beyond Jamie MacDonald.