The home side completely dominated the Scottish Premier League match but could not force the ball over the line until Sutton's late intervention.
Gary Harkins made his second full debut in a Dundee shirt, playing in the hole behind Colin Nish, but it was the defensive side of Barry Smith's team that deserved the praise after coming so close to recording their third consecutive clean sheet against the Edinburgh side.
Hearts manager John McGlynn had started with a youthful side for the Scottish Communities League Cup semi-final win against Inverness and, buoyed by the vindication of that decision, he decided to go one further, granting teenager Billy King his first-team debut in place of Andrew Driver.
And the young attacker had the first opportunity within three minutes. Slack play in the back-line allowed the loose ball to fall to King who cut in and scuffed a shot wide from 18 yards.
It was King's team-mate on the opposite wing who created the next opportunity. Leading a counter-attack, Jamie Walker drove into the box before squaring for Michael Ngoo. His shot from close range was blocked on the line by Gary Irvine, with the ball deflecting back into the path of Walker whose toe-poke was held by Rab Douglas.
On the half-hour Ngoo demonstrated his strengths by controlling a difficult ball, muscling his way into position and letting fly with a powerful drive that was deflected wide by Declan Gallagher.
Five minutes later it was the turn of Douglas to keep the score level, pulling off an outstanding finger-tip save to deny Walker, while Ngoo then saw his goal-bound header deflect off Colin Nish and land on top of the crossbar.
The woodwork was also rattled in the second half, Jason Holt was the perpetrator this time, looping a shot from 25 yards onto the bar.
Hearts, with so many of their players having slogged through 120 minutes of football four days previous, tired in the latter stages with McGlynn unwilling to make a change until there was little more than five minutes of the match remaining.
And it was the two players introduced that finally made the breakthrough. Groans echoed around Gorgie when Arvydas Novikovas let a cross-field pass from Walker slip under his foot. However, the Lithuanian recovered and sent in a deep curling cross that was powered into the net by Sutton at the far post.
Two minutes later Novikovas had his own chance to get on the scoresheet but his sliding effort from Ngoo's cross went just wide.