Warren Joyce's side had to wait to make the breakthrough but late strikes by Sean McGinty and John Cofie were enough to take the three points.
A tight encounter ensued at The Hawthorns with United still limited in terms of attacking options at this level. Adnan Januzaj was asked to push forward and went through after only three minutes, latching onto Marnick Vermijl's long ball and beating Boaz Myhill, only for Smart to hack clear.
Myhill watched a long-ranger from McGinty fly over the top after 28 minutes but the best chance of the half fell to Januzaj again, as the Belgian's shot brought the best out of Myhill following a Tyler Blackett cross from the left.
The second half was equally competitive until the Reds came on strong late on. Januzaj was unlucky not to win a 65th-minute penalty when the referee reversed his original decision and Charni Ekangamene kept his composure inside the box to crack a curler against the bar as the pressure started to build.
The impressive Januzaj then got to the byline to cross for McGinty to hammer home with only six minutes left to finally break the deadlock.
Moments later, Anders Lindegaard's punt forward was missed by the defence and substitute Cofie was in the right place at the right time to place the ball wide of Myhill with his right foot.
United: Lindegaard; Vermijl, Fornasier, McGinty, Blackett (Cofie 68); Thorpe, Veseli, Cole, Ekangamene, Lingard (McCullough 90); Januzaj (Lawrence 88). Subs not used: Gollini, Daehli.