Demba Ba had given the Magpies the lead, prodding home the rebound after seeing his penalty saved, before Ruben Amorim equalised for Braga.
But Taylor won the trophy for Alan Pardew's men when he curled home a delightful 25-yard free-kick past the helpless Beto.
Pardew had named a strong side with Senegal strikers Ba and Papiss Cisse both featuring from the start.
Despite their presence, it was an uneventful first half with Cisse squandering the best opportunity as he fired over following good work from Danny Simpson.
But a Newcastle side also featuring Cheick Tiote and Fabricio Coloccini took the lead after the break when Sylvain Marveaux was tripped inside the area.
And after Ba netted at the second attempt only for Amorim to level things up, it was Taylor who secured a morale-boosting victory.
Harper; Simpson, Tavernier, Coloccini, R.Taylor; Sammy Ameobi (Obertan 88), Tiote (Cabaye 70), Gutierrez, Marveaux; Cisse, Ba (Vuckic 85).