Brandy struck his first of the afternoon on 24 minutes by rounding Magpies goalkeeper Bartosz Bialkowski before slotting into the empty net.
Nine minutes later the 24-year-old doubled his team's advantage, expertly smashing past Bialkowski from inside the area after Craig Westcarr had gone close.
The Saddlers looked to capitalise on the punch-drunk state of their opponents as Westcarr had the opportunity to kill off the contest shortly after, but the Polish shot-stopper produced a smart save to maintain the two-goal deficit.
However, Brandy secured the matchball moments before the interval by getting on the end of Westcarr's cross to head his team's third.
Walsall picked up where they left off as Andy Butler fired the visitors into a 4-0 lead just four minutes into the second half, before Westcarr got his name on the scoresheet shortly after.
Callum McGregor hit a free kick mi-way through the half to reduce the arrears, but it proved little more than a consolation as the visitors climb into the playoff places in the League One table at the expense of Peterborough United.