Paul McGuinnness' side created plenty of chances throughout the 90 minutes at Bodymoor Heath and could have won by a bigger margin.
The Reds had to wait until the 50th minute for the deadlock to be broken by Weir who netted his first Academy league goal for the club.
Adnan Januzaj's dangerous cross from the left wing was met by James Wilson in the box. His first-time effort was well saved but Weir was on hand to slot home the follow-up from close range.
Substitute Jack Barmby, who had replaced Mats Daehli at the break, doubled the advantage on 70 minutes from the penalty spot.
Barmby tricked and twisted his way towards goal before he was tripped in the box. He dusted himself down and stepped up to take the spot-kick which, despite slipping as he struck the ball, he tucked away to make it 2-0. And that's how it finished despite the Reds creating more opportunities thereafter.
Next up for United is a trip to Sunderland who put six past Newcastle on Saturday.
United: Sutherland; Love, Grimshaw, Dalley, Rowley; Weir (Gorre 70), Pearson, Rudge, Januzaj, Daehli (Barmby 46); Wilson (Byrne 68). Subs not used: Gollini, Evans.