Hearts displayed the pleasing knack of picking up three points without playing well.
Goals from summer signings Jean Louis Valois and Mark De Vries were enough to take this game out of the reach of Dundee United.
But the visitors had more possession and more attempts at goal but their finishing was woeful at times.
Charlie Miller missed a couple of first-half chances and United finished strongly but their misery was complete shortly before the end when substitute Allan Smart was carried off with ankle ligament damage.
United were first to threaten, but Steve Thompson shot straight at goalkeeper Roddy McKenzie from 16 yards.
At the other end, a cross from Phil Stamp found its way to De Vries but the Dutch striker's shot lacked conviction and Paul Gallacher saved comfortably.
Miller should have scored after 22 minutes after being put through by Derek Lilley, but he slid his shot wide with only McKenzie to beat.
It was the warning Hearts needed and they stepped up the pace. A Scott Severin cross found De Vries on his own in the penalty area but he completely missed his header.
The Dutchman's blushes were spared moments later when Valois shot Hearts ahead. De Vries miscued as he attempted to get on the end of a Steven Boyack cross and Valois followed up to steer the ball home.
Miller sliced another shot wide as United tried to get back into the game.
But it was Hearts who always looked the more deadly in front of goal. Andy Kirk sidestepped Gallacher, but found only the sidenetting with an angled shot in the 50th minute, and three minutes later, United came close to the equaliser.
Miller's corner was headed on by Thompson, but Stamp headed the ball off the line.
Hearts scored the clinching second goal three minutes later. De Vries was left unmarked at a Boyack corner and his volley had enough weight to carry it over the line as United sought desperately to clear.
Gallacher made a brilliant save midway through the second half to keep out a Kirk header and, though United dominated the closing stages, their finishing was poor