Following a dominant first half, Asante gave Solihull the lead five minutes after the break, scrambling home after a long throw by Connor Franklin.
It was 2-0 on 70 minutes, when Omari Sterling-James' tame shot from the right-hand side somehow found its way past Magnus Norman after the winger linked up with Jamey Osbourne.
Asante, who also scored a hat-trick against Maidstone on Saturday, wrapped up the points with two goals in the last 20 minutes.
First he latched onto Ryan Beswick's pass to finish well before completing the rout eight minutes from time after peeling away from the Southport backline.
Report supplied by the Press Association
Match ends, Solihull Moors 4, Southport 0.
Second Half ends, Solihull Moors 4, Southport 0.
Substitution, Solihull Moors. Darryl Knights replaces Shepherd Murombedzi.
Goal!  Solihull Moors 4, Southport 0. Akwasi Asante (Solihull Moors).
Substitution, Southport. Declan Weeks replaces Jordan Lussey.
Shepherd Murombedzi (Solihull Moors) is shown the yellow card.
Goal!  Solihull Moors 3, Southport 0. Akwasi Asante (Solihull Moors).
Goal!  Solihull Moors 2, Southport 0. Omari Sterling-James (Solihull Moors).
Substitution, Southport. Bobby James Moseley replaces James Caton.
Ryan Higgins (Southport) is shown the yellow card.
Substitution, Southport. James Gray replaces John Cofie.
Andrai Jones (Southport) is shown the yellow card.
Goal!  Solihull Moors 1, Southport 0. Akwasi Asante (Solihull Moors).
Second Half begins Solihull Moors 0, Southport 0.
First Half ends, Solihull Moors 0, Southport 0.
First Half begins.
Lineups are announced and players are warming up.
A second-half hat-trick from the in-form Akwasi Asante gave Solihull Moors back-to-back wins for the first time this season as they brushed aside Southport.