Porto, Champions League winners ten years ago under Jose Mourinho, have quickly been revitalised by former Spain under-21 coach Julen Lopetegui, winning six and drawing one of their seven matches this season.
Lopetegui was brought in after a relatively poor last season when Porto finished third in the league and fell in the Champions League group stage.
Brahimi, making his Champions League group debut, took full advantage of BATE's generosity, starting with a gift from BATE goalkeeper Sergei Chernik in only the fifth minute.
He miscued a throw straight to Brahimi, who ran into the penalty area and hit the ball high into the net from a narrow angle.
Brahimi, who represented France at all age levels from Under-16 to Under-21 before declaring for Algeria, added a fine solo goal after 32 minutes, running from the halfway line and veering inside before shooting in off the far post.
As Porto's dominance continued, Colombia forward Jackson Martinez volleyed against a post before scoring the third in the 37th minute.
Martinez also headed against the bar early in the second half but the home crowd did not have to wait long for further goals.
Brahimi curled in a superb free-kick for his hat-trick after 57 minutes and four minutes later substitute Adrian Lopez, who appeared in last season's final for Atletico Madrid, scored the fifth.
Vincent Aboubakar, substituting for Martinez, completed the rout with quarter of an hour left as Porto surpassed their previous biggest victory in the competition, a 5-0 away success against Werder Bremen 20 years ago.
"For a first match it is incredible," Brahimi, who helped Algeria reach the second round at the World Cup, said after the first hat-trick of his career.
"This was also the first time I scored three goals. We played very well and we must go on like this. I’m very happy. I have to thank all."
(Writing by Steve Tongue; editing by Martyn Herman and Brian Homewood)
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