Lee Carsley gave Blues a 1-0 lead just moments before half time and Marcus Bent doubled their advantage just five minutes after the break.
Blues took a 3-0 lead in the eightieth minute when Brice Ntambwe headed home a Robin Shroot corner.
The home side scored what only proved to be a consolation goal two minutes from the end of normal time through Ross Dempster.
Solihull Moors 1 Blues 3 ..
Solihull Moors: Crane, Midworth, Campion, Streete (Stapleton, 52), Khela, Faulds, English, Pugh, Gardner (Lea, 60), McPike (Dempster, 70), Mills. Subs not used: Price, Douglas.
Blues: Butland, Preston, Metcalfe, Ridgewell (Hubbins, 60), Jacob Rowe, Ntambwe, Hronec, Carsley (Gombala, 60), Bent (Shroot, 60), Bogle, Williams.
Subs not used: Townsend, Fleet.