Jackie McNamara's men have now won five of their last six league matches to close the gap on Aberdeen and Motherwell, who are both in action on Saturday.
Hearts' 19th defeat of the campaign keeps them 21 points behind second-bottom St Mirren, with relegation from the top flight seemingly a formality for the Edinburgh club.
The visitors opened the scoring on 35 minutes. Winger Stuart Armstrong cut in from the left before playing a neat reverse pass into Andrew Robertson, who in turn rolled the ball across the six-yard box for Brian Graham to tap home.
Callum Paterson spurned a chance to pull Hearts level before Nadir Ciftci doubled United's lead in the 70th minute.
Having cut in from the right, the striker worked a shooting opportunity and lashed the ball into the far corner.
Hearts halved their deficit on 78 minutes when Danny Wilson headed home from a Sam Nicholson corner, but it was too little, too late for the league's bottom club.
