Not surprisingly, United manager Jim Goodwin had made just one change, sticking with the second-half line-up that came from behind to beat Celtic on Wednesday.
However, Ryan Strain was forced off through injury after just five minutes and replaced on the right of defence with Dario Naamo.
Hibs' first chance came from that area of the park as Nicky Cadden sliced a thundering drive into the side-netting.
However, it was United who took the lead as Will Ferry knocked the ball back into the path of Stephenson and the on-loan Liverpool midfielder prodded home through a sea of legs.
Hibs remained the more dangerous side and, after Kieron Bowie had fired just wide, Boyle powered a drive home at the back post from a corner.
Debutant on-loan Charlton Athletic centre-back Zach Mitchell headed against the crossbar as United goalkeeper Dave Richards was left sporting a cut eye, while United's Will Ferry did likewise at the other end direct from a corner.
United have made a habit of improving after the break in recent weeks and so it was again, but substitute winger Kristijan Trapanovski was denied by goalkeeper Raphael Sallinger and also fired just wide.
At the other end, replacement striker Thibault Klidje's shot was half-stopped by the hand of goalkeeper Richards before being hacked clear.