After a first half desperately low on class, Mexico midfielder Hector Herrera lashed Porto into a 56th-minute lead with a stunning goal.
That sparked the Portuguese outfit into action and they were soon two goals to the good, as Jackson Martinez swept his sixth goal of this season's competition home before substitute Cristian Tello netted a late third.
Barcelona loanee Tello's effort left BATE further red faced as they claimed a share of an unwanted Champions League record.
BATE have now conceded 22 goals in their five Group H games this season - equalling the record shared by Dinamo Zagreb and Nordsjaelland, with Athletic Bilbao to come in their final game.
The 2004 champions Porto, meanwhile, look in good form and could need just a point when Shakhtar Donetsk visit the Estadio Dragao on matchday six to guarantee top spot and potentially avoid some of Europe's elite in the last 16.
BATE came into the match needing three points in order to keep their slim hopes of progression alive but rarely looked capable of getting through the visiting defence in a lacklustre opening to the contest.
Aleksandr Karnitskiy wastefully launched a 30-yard free-kick into the stands to sum up a turgid opening half-hour bereft of any clear goalscoring chances.
Having already qualified, Porto's more patient and conservative approach was perhaps understandable, though coach Julen Lopetegui grew visibly irked by his side's half-paced play as the opening period wore on.
Yacine Brahimi - scorer of a hat-trick in Porto's 6-0 demolition of the Belarusians in September - curled a set-piece narrowly wide of Sergey Chernik's left-hand post as the sides deservedly headed in at the break goalless.
However, the game was lit up by its first moment of genuine quality as Porto took the lead in stunning fashion 11 minutes into the second period.
Casemiro robbed Karnitskiy of possession in his own half before laying the ball into the path of Herrera, who unleashed a swerving effort from the right-hand angle of the penalty area which beat a bewildered Chernik.
Though less spectacular, Porto's second was still neatly constructed - with Herrera again involved.
Brahimi burst down the left flank before playing the ball to the Mexican in the penalty area, who held off his man and rolled into the path of Martinez to stroke a shot into the far corner.
The damage was done, but there was still time for substitute Tello to weave through the defence to score a third with a minute to go.
That left BATE requiring a first clean sheet in the competition proper since their December 2008 0-0 draw at Juventus if they are to avoid going down in history for the wrong reasons.
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