The Spanish striker broke the deadlock at the end of a scrappy opening period when he calmly converted a penalty kick to put Valencia ahead.
BATE came out fighting in the second half but were undone by an incisive counter-attack after 60 minutes.
Andre Guardado burst forward down the wing before picking out Soldado with a pinpoint cross. The in-form striker made no mistake with a rasping volley, giving BATE keeper Andrei Gorbunov no chance.
Soldado completed a brilliant hat-trick 10 minutes later, exchanging passes with Tino Costa before coolly fishing from close range for his second Champions League treble.
An emphatic victory moves Valencia level with BATE and Bayern Munich on six points, leaving Group F finely poised at the halfway stage.