Dale Hilson was the matchwinner at Riccarton, scoring the only goal of a proverbial game of two halves.
And the three points takes United back up to joint top of the Youth League, albeit temporarily.
Without Scott Allan because of illness and Ross Smith who is still sidelined with injury, United were slow to start.
Indeed, Hearts had by far the better of the first half, and the visitors were fortunate to still be level at the break.
That was only thanks to one quite outstanding save by Marc McCallum from Walker.
In addition, the home side missed two other very good chances.
The closest chance by United was a shot from Rayn Dow that was saved by Lyness.
However, with Tannadice coach Stevie Campbell making a couple of tactical switches at the interval, the second half was a different story with keeper McCallum virtually an onlooker.
Craig Hill went close with a header, but it was Dale Hilson with his 13th goal of the season that earned his side victory.
It came 15 minutes from the end when Davie Shaw got his head to a corner and Hilson was at the back post to apply the finishing touch.
Dundee United: McCallum, Irvine, Mosson, Hill, Shaw, Stokes, Scott, Lee, Hilson, Armstrong, Dow.
Subs not used: L Swanson, Murray, W Robertson, Thomson. Elfverston.