Sho-Silva's brace came either side of Jordan Higgs' strike to give the hosts a 3-0 lead just after half-time.
Jordan White began the visitors' comeback before Ntumba Massanka got his on the scoresheet on the hour mark.
Blair Turgott claimed Bromley's fourth before Massanka's second for Wrexham, who dropped from 10th to 12th.
Match ends, Bromley 4, Wrexham 3.
Second Half ends, Bromley 4, Wrexham 3.
Shabazz Omofe (Bromley) is shown the yellow card.
Substitution, Bromley. Shabazz Omofe replaces Jordan Higgs.
Goal!  Bromley 4, Wrexham 3. Ntumba Massanka (Wrexham).
Goal!  Bromley 4, Wrexham 2. Blair Turgott (Bromley).
Substitution, Wrexham. Leo Smith replaces Iffy Allen.
Martin Riley (Wrexham) is shown the yellow card.
Substitution, Bromley. Bradley Goldberg replaces Tobi Sho-Silva.
Substitution, Bromley. Lee Minshull replaces Connor Dymond.
Goal!  Bromley 3, Wrexham 2. Ntumba Massanka (Wrexham).
Substitution, Wrexham. Ntumba Massanka replaces Izale McLeod.
Goal!  Bromley 3, Wrexham 1. Jordan White (Wrexham).
Goal!  Bromley 3, Wrexham 0. Tobi Sho-Silva (Bromley).
Second Half begins Bromley 2, Wrexham 0.
First Half ends, Bromley 2, Wrexham 0.
Goal!  Bromley 2, Wrexham 0. Jordan Higgs (Bromley).
Goal!  Bromley 1, Wrexham 0. Tobi Sho-Silva (Bromley).
First Half begins.
Lineups are announced and players are warming up.
Tobi Sho-Silva struck twice as Bromley held out against Wrexham's fightback in the National League to rise three places to 11th in the National League.