At the end of a turbulent week for the Edinburgh outfit, Robertson dived to head in the only goal of Saturday's SPL game at Tannadice in the 24th minute to capitalise on a magnificent pin-point cross from home skipper Jon Daly.
Rudi Skacel was denied an equaliser by an offside flag only seconds later and Daly saw a second-half strike ruled out by the same linesman, but Robertson was to prove the match-winner.
The Hearts players had this week been hit with a delay in their wages for the second consecutive month, with owner Vladimir Romanov sparking further fears over the club's future by reiterating his desire to sell up.
However, with manager Paulo Sergio forced to watch from the stand because of a five-match touchline ban handed down by the SFA this week, there was to be no happy conclusion to a worrying week.
Hearts had not won any of their last three matches since the wages issue blew up last month, when the first-team squad had to wait 19 days to receive their salaries, and this defeat stretched that run to four.
Sergio, without suspended duo Ian Black and Danny Grainger, pulled off a surprise by bringing John Sutton back in from the cold for his first SPL start since August 13, at the expense of Mehdi Taouil, but the former Motherwell marksman was ineffectual as the lone striker.
There was a familiar face for the visitors in the United starting line-up, with former full-back Robbie Neilson being handed his debut for the hosts after this week signing until the end of the season.
All eyes were on how easily Hearts would be able to shrug aside their off-field problems and they made a bright opening in which David Templeton came within a whisker of breaking the deadlock in only the ninth minute. Latching on to a loose ball on the edge of the area, the 22-year-old's deliberate, curling effort slipped inches wide of the post with United keeper Dusan Pernis diving at full stretch to his left.
But it was the home side who eventually took the lead midway through the half with a superb goal that was all down to a wonderful assist from Daly. The Irishman peeled off his marker wide on the right and sent in a magnificent curling cross which evaded Andy Webster in the middle and was gleefully headed in by the diving figure of Robertson from 12 yards out.
Hearts thought they had come up with the perfect riposte only a minute later, but Skacel's strike after Templeton's run to the heart of the United defence was ruled out by an offside decision. Tynecastle boss Sergio, watching from his seat at the back of the Tannadice main stand, would have been heartened by his team's response to going a goal down, as they continued to look a threat going forward.
Full-back Ryan McGowan tested Pernis with a left-foot cross which the United keeper was forced to claw away from under his crossbar in the 35th minute.
Two minutes later, the Slovakian held Templeton's cross-cum-shot before he beat away Skacel's right-foot drive at the climax of a lightning break.
Hearts continued to press after the interval but Sutton brought jeers from the home support with one wayward effort from inside the box and Skacel wasted a free-kick when he curled his set-piece shot high and wide from 22 yards out in the 51st minute.
The Czech midfielder at least got his effort on target two minutes later from Eggert Jonsson's left-wing cross but Gavin Gunning did enough to partially block and Pernis held comfortably. Hearts kept up the pressure but Templeton could not seriously trouble Pernis with a low drive at the end of another mazy run in the 57th minute.
The Tynecastle outfit made their first change after an hour's play when they withdrew defensive midfielder Adrian Mrowiec in favour of the introduction of the attack-minded Mehdi Taouil.
Taouil's trickery created a chance only two minutes later but Sutton mis-hit his low shot into the arms of Pernis after enjoying the break of the ball 10 yards out. United then almost made Hearts pay for their lack of cutting edge when they had the ball in the net in the 67th minute.
However, Daly was flagged for offside after Kello's save from Gary Mackay-Steven's powerful drive fell kindly to him eight yards out. United were in danger of sitting too deep as time wore on.
Hearts certainly showed a willingness to chase the game, throwing on striker Stephen Elliott and winger Arvydas Novikovas for Sutton and full-back Jamie Hamill.
However, despite piling men forward in the closing stages when they enjoyed a stranglehold on possession, the United defence held firm to seal three hard-earned points.