Superb Stuart Findlay netted deep in second-half injury time to claim his fourth goal in five appearances for Hearts and secure three hard-earned points for the Men in Maroon at Tannadice.
The Tynecastle side top the William Hill Premiership table on six points, the same as Premiership title-holders, Celtic, who are second, but it took nearly 95 minutes to get the better of a well-disciplined Dundee side.
Earlier, Hearts showed one change from the 2-0 win over Aberdeen at Tynecastle on Monday, Christian Borchgrevink missing out through injury and the Norwegian is likely to be sidelined for a number of weeks.
Alan Forrest got the nod from Derek McInnes against a Tayside team on a high after their 2-2 draw at Rapid Vienna in the Conference League in midweek.
Burkina Faso international, Pierre Landry Kabore, was on the bench after putting pen to paper on a three-year deal and Hearts started brightly.
Blair Spittal and Harry Milne combined well on the left, but the ball was blocked, then Claudio Braga threatened, forcing a corner from which Frankie Kent had a real chance, lying unmarked in the box, but the big defender headed well over, a great chance missed.
At the other end, Hearts cleared after a long throw on the United left, but it was pin-ball as the ball bounced around, but, seconds later defender Vicko Sevelj forced Zander Clark to make a diving save to his right with a shot from the edge of the box.
The Jambos patiently stroked the ball around the lush turf looking for an opening and it came when Forrest stroked a left-foot cross towards the United six-yard box and Braga broke free only to send the ball straight into the gloves of Yevhen Kucherenko. Another good chance gone.
Milne then forced a diving save from Kucherenko as Hearts continued to press and they earned a corner on the left. Milne’s kick was floated to the far post and Stuart Findlay was hauled to the ground by Lurie Iovu.
Lawrence Shankland took responsibility and he took two steps before placing the ball with his instep to the right as the goalkeeper dived in the other direction. Hearts one up and time 19 minutes.
Two minutes later the sides were level. Kent fouled Dundee defender Owen Stirton 25-yards out and Croatian Ivan Dolcek, a summer loan signing by Jim Goodwin, sent his left-foot free-kick high into the rigging from a central position and just outside the penalty box. The ball evaded Clark’s dive.
Once again, the Men in Maroon dominated possession but were unable to penetrate the United rearguard until Milne’s persistence won a free-kick near the left corner flag but the ball was hooked clear.
It was recycled with Braga twice threatening to break clear. United cleared the danger to relieve the pressure before Milne was booked after 31 minutes, the first yellow card of the game, apparently for kicking the ball away.
Surprisingly, United defended deep, allowing Hearts to camp out in their half, but the home side were compact and difficult to penetrate, and they broke forward. Zak Sapsford drove into the box. Halkett was there to block.
Seconds later United were in front and Dolcek was the man to make the visitors pay, running onto a defence-splitting pass to smash the ball at Clark who allowed it to squirm past him and into the net. The goalkeeper held his head in his hands. Time 42 minutes.
Shankland had a chance at the back post following a cross from Findlay but he headed wide and United threatened again through Sapsford but this time Clark booted clear and it was promoted to the other end and Shankland set-up Braga whose left-foot shot was saved by Kucherenko.
It was the Jambos last opportunity of the half and United marched back to the dressing room in a high, leading 2-1.
Forrest and Kent made way for Michael Steinwender and Alexandros Kyziridis being introduced as McInnes looked to level. United stayed tight at the back, forcing Hearts to knock the ball around looking for a breakthrough.
They lacked penetration as disciplined United maintained their slender lead, but Camara was booked for a foul on Osian McEntee. Spittal took the free-kick but the ball flew high and wide.
The near 3,000 Jambos fans in the crowd of over 10,000 grew more and more frustrated at the lack of progress from their favourites, with United keeping Hearts at bay, forcing the visitors wide, and increasing the frustration of the visiting players who became sloppy.
Kyziridis and Milne were involved in an neat interchange which earned a fifth corner. Milne sent the ball to near the penalty spot and Findlay, moving backwards, expertly controlled his header back across goal and into the net. A great finish and Hearts were level at 58min. Game on.
Devlin then found Shankland and the skipper fired towards goal but the pace was taken off the ball by a defender and the United goalkeeper collected comfortably.
United broke and Clark made a hash of a clearance kick, finding a United player, but luckily he was sloppy with his finish.
McInnes had seen enough and he introduced Sabah Kerjota for Spittal in a bid to edge Hearts ahead against a side whose energy levels were being tested after their exertions in Europe on Thursday.
Goodwin then made two changes, Camara and Sapsford making way, and Amar Fatah and Kai Fotheringham coming on.
United goalkeeper Kucherenko needed attending for a hand injury but continued and Hearts frustration grew by the minute. Too many aimless balls were fired into the United defence and were easily dealt with, the attacks needed more pace and the wall of orange shirts continued to soak up the pressure.
McInnes decided on another change, Milne retreating and Stephen Kingsley coming on for the final 13 minutes of regulation time as United broke causing a few anxious moments in the Jambos box.
A spark was required from a man in maroon but Braga was shown a yellow card for a foul before McInnes called on James Wilson to replace him with seven minutes remaining. Wilson, of course, was red carded in his last appearance against United in April.
Then Kingsley and Sevelj needed treatment after a clash of heads before Kerjota tried to thread the ball through the United defence without success in a stop-start game and it was the home side who came close with Sevelj firing inches wide. A late scare for Hearts.
But it was the Jambos who came closes with a shot from Kerjota saved by Kucherenko then Steinwender had a header saved on the line and scrambled for a corner.
Kyziridis took one for the team with a late yellow as United threatened to break upfield and it was Hearts who showed more urgency and won a corner on the left.
It was Greek winger Kyziridis who swung the ball in and 29-year-old Findlay, on loan from EPL Championship side, Oxford United, ran from the penalty spot to stoop low at the near post to stun United and end their defiance in the third minute of injury time.
HEAD MAN: Stuart Findlay (centre) claims the game-winner for Hearts. Picture Nigel Duncan
