Mauricio Pellegrino's side have had a poor start to the season and were on the back foot against the surprise early group pace-setters, until Soldado netted from the penalty spot just before the break.
The visitors ran away with the game in the second half, when they could have won by a much bigger margin, and seized first place from the Belarus champions on six points but with a superior goal difference to BATE and Bayern Munich.
Soldado had an early shot saved, but it was BATE who took control early on showing the confidence born from having shocked Lille and last year's finalists Bayern by the same 3-1 scoreline in their opening games.
Visiting goalkeeper Diego Alves had to be alert to keep out a couple of dangerous balls and was lucky Vitali Rodionov's point-blank diving header was directed straight at him in the 38th minute.
Soldado won himself a penalty just before half-time, tumbling to the ground as Marko Simic tugged on his arm, and he stroked the ball home from the spot.
Valencia's second came from a counter-attack when Andres Guardado crossed and Soldado volleyed home after 55 minutes, and a clever one-two in the area with Tino Costa allowed the Spain striker to bag his third in the 69th.
The result is a boost to Valencia as they recorded their first away win in 12 matches in all competitions.
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