Scotland midfielder Paul Hartley scored from the penalty spot as Hearts survived late Dundee United pressure to take all three points at Tannadice.
Paul Hartley celebrated with missing captain Craig Gordon
United goalkeeper Derek Stillie made a couple of fine saves, while team-mate Craig Conway had a horrendous miss.
Hartley fired home after Stuart Duff handled inside the box.
Barry Robson had shots denied by the crossbar and Hearts goalkeeper Steve Banks, while Garry Kenneth had another effort cleared off the line.
Banks had been a surprise inclusion after the ommission from the starting line-up of Scotland international Craig Gordon.
Head coach Valdas Ivanauskas insisted that illness rather than a midweek disciplinary meeting was the reason why the 23-year-old was on the bench.
Defender Christophe Berra took over the captaincy, while Robbie Neilson, Michal Pospisil, Andrew Driver and Hartley all returned to the starting line-up.
Steven Robb, David Goodwillie, Kenneth and Duff started for Dundee United in place of David McCracken, Alan Archibald, the suspended Christian Kalvenes and the injured Noel Hunt.
And the reshuffled United side took only three minutes to draw Banks into the action, the Englishman parrying a dipping, 25-yard drive from David Robertson that was then cleared by Neilson.
Hearts started to control the midfield area, led by the inspirational Hartley, although Banks had to do well to gather a low drive from Robson.
Stillie did even better in the other goal, with a fantastic save after Marius Zaliukas clipped a right-footed shot towards goal following a Hartley free-kick.
Craig Gordon had spent the afternoon on the Hearts bench
Hearts almost fell behind when Collin Samuel set up Conway, but the midfielder placed the ball over the bar from a couple of yards out.
United lost Robb to injury in the 33rd minute, with McCracken coming into the centre of the defence and Mair switching to left-back.
Andrius Velicka and Bruno Aguiar both squandered opportunities from six yards, but it was goalless at half-time.
Stillie did well to block immediately after the restart as Velicka missed another chance from close range.
Hearts were awarded a penalty after 53 minutes, when the ball appeared to strike the hand of Duff.
Hartley made no mistake from the spot, blasting the ball past Stillie, and the Scotland midfielder set up Michel Pospisil inside the box only for the Czech striker to fire into the side-netting.
Hearts survived a Robson free-kick that blasted off the crossbar and Banks, moments after receiving treatment for an injury, produced a fine save at full stretch following another Robson set-piece.
Berra headed a Kenneth effort off the line, but Hearts claimed the victory.
Dundee Utd: Stillie, Duff, Kenneth, Mair, Robb (McCracken 34), Conway, Kerr, Robertson, Robson, Samuel, Goodwillie (Smith 68).
Subs Not Used: McLean, Archibald, Cameron, Burnett, Watson.
Booked: Mair, Robertson.
Hearts: Banks, Neilson, Berra, Zaliukas, Fyssas, Mikoliunas (Barasa 51), Hartley, Aguiar (Brellier 85), Driver, Velicka (Jankauskas 71), Pospisil. Subs Not Used: Gordon, McCann, Bednar, Wallace.
Booked: Pospisil, Barasa, Aguiar, Zaliukas.
Goal: Hartley 55 pen.
Ref: C Richmond