Brown's first came in the 38th minute when Peter Lovenkrands' attempted lob was kept in by Amari'i Bell allowing the midfielder to tap home from close range.
Striker Reece Hales, on as a substitute, doubled Blues' advantage in the 67th minute after reacting to a quickly-taken free-kick over the top to lob the hosts' goalkeeper shortly after Solihull Moors had made a raft of substitutions.
Brown bagged his second two minutes later and saved his best until last by firing home a right-footed drive from 25-yards into the bottom left corner.
Non-league Solihull, of the Conference North, pegged a goal back with four minutes remaining courtesy of substitute Trey Braithwaite's smartly-taken free-kick from 25 yards.
Solihull Moors starting XI: Singh, Nottingham, Gough, Fleet, Elvins, Pierpoint, Till, Beswick, Bogle, Birley, Knights. Subs: Connolly, Martinez, Denny, Kemp, Ballinger, Grandison, Hayden, Taylor, Birch, Wint, Brathwaite.
Blues Development XI: Preston, Adams, Painter, Truslove, McDonald, Fry, Bell, Brown, Obeng, Gray, Lovenkrands. Subs: Hales, Preston, Quirk, Henderson, Kalenda, Asamoah.