Villa took control after the restart and went ahead with a soft penalty, awarded on 52 minutes when Graham went to ground as Reece James tried to pull him back. Graham wasted little time in sending Sutherland the wrong way from the spot.
United failed to respond to the setback and Joyce threw on Mats Daehli for Lawrence but the Midlanders looked in control with Graham forcing a simple save out of Sutherland after Ekangamene was robbed.
Mason Watkins-Clark saw a flying header blocked and Jack Grealish shot wastefully wide after being allowed to skip between Frederic Veseli and Bebe. The Reds rarely looked like salvaging a point, despite one dangerous centre from Bebe, although Veseli hammered a volley over the top towards the end with the only real attempt of the second half.
The result, a rare home defeat - United's first for over two years, takes the Midlanders above Joyce's team in the table and means the pressure is still on in the bid to qualify for the elite stages.
United: Sutherland; Veseli, McCullough, Fornasier, James; Bebe, Ekangamene (Grimshaw 74), Hendrie (Barmby 76), Lawrence (Daehli 58), Januzaj; Henriquez. Subs not used: Gollini, Wilson.