Augmented by first teamers Ryan Gauld and John Souttar, the Tannadice teenagers proved too good for their Tynecastle counterparts in a hard fought encounter at Newtongrange.
Hearts made the brighter start with Jamie Walker and Angus Beith both firing narrowly past.
On 13 minutes, Scott Fraser drilled a free-kick inches past as United took control.
And three minutes later, the Tangerines took the lead. Liam Gordon impeded Aidan Connolly in the box and the winger got up to convert the resultant penalty.
There was a rare shot on target after 21 minutes from United's Rafa Malaga but the Maroons's keeper saved comfortably.
With half an hour gone Brad Smith curled a shot which brushed the side-netting but the rest of the half produced no further goal attempts from either side.
Continuing to be the more fluent side, Ryan Gauld and then Malaga went close.
But Hearts' first shot on target after 63 minutes almost brought them a goal, United keeper Joe McGovern brilliantly blocking a Gary Oliver effort.
This sparked a spell of home dominance, but United stood firm.
And with quarter of an hour left only a superb one-handed save by Mark Ridgers prevented Gauld doubling the lead from a free-kick.
However, it was only a temporary reprieve, because from the corner, Adam Harwood bundled the ball in to put the game beyond the Jambos.
Dundee United: McGovern, Montgomery, Ballantyne, S Smith, J Souttar, Harwood, B Smith, Gauld, Malaga (Tumilty 66), Fraser, Connolly.
Substitutes not used: Anderson, Gilmour, Tumilty, Allan.