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The Portuguese side took the lead in the fifth minute when the visitors' keeper Sergey Chernik threw the ball out of the box straight into the path of Brahimi, and he made no mistake firing it into the top corner.
BATE almost responded immediately when Mikhail Gordeichuk showed a great burst of pace to break from midfield and he got into the box. However, Fabiano raced out of his goal to narrow the angle and make the stop with his feet.
Jackson Martinez should have doubled the home side's lead when he latched on to a ball from Brahimi at the back post, but his chipped effort smacked off the post and bounced clear.
Brahimi did double his side's lead with an individual goal as he raced the length of the pitch and weaved his way inside the box before powering an effort past Chernik into the net.
Porto continued their dominance as they netted for a third time when Danilo beat a poor challenge down the right and whipped a dangerous cross to the near post, which was converted by Martinez.
After the break, Martinez nearly added to his tally for the evening as he met Ricardo Quaresma's cross into the middle but his effort clattered off the crossbar and went over.
Brahimi scored his hat-trick and his side's fourth goal of the night when his side won a free kick just outside the penalty area. The forward stepped up and curled the set piece into the top corner.
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Adrian Lopez added a fifth goal for the Portuguese side when he raided forward and lifted the ball over Chernik with a neat finish, while Vincent Aboubakar wrapped up the win for the home side as they broke on the counter and the striker fired the ball into the net.