The 27-time Portuguese champions claimed just one home point from a possible nine before failing to reach the knockout stages in last season's competition, but an early opener and a superb second from the Algeria winger put them firmly in control at the Estadio do Dragao.
Jackson Martinez, with his 67th goal on his 101st Porto appearance, made it 3-0 with a bullet header before the break and Brahimi completed his treble when he curled home a brilliant free-kick before going off to a standing ovation after 59 minutes.
Adrian Lopez and Vincent Aboubakar scored their first goals since joining from Atletico Madrid and Lorient respectively, ensuring that Porto comfortably recorded their biggest Champions League victory in what was their 200th appearance in Europe's elite club competition.
Julen Lopetegui's side have now also kept clean sheets in six of their past seven outings in all competitions.
Brahimi made it 2-0 just after the half-hour mark – and in electrifying fashion.
The 24-year-old, signed from Granada for just a €6.5 million after he impressed at the FIFA World Cup, collected possession in his own half, sprinted past one challenge to the edge of the box, cut inside another and unleashed a fierce, low drive that went in off the inside of the post.
It got worse for BATE eight minutes before half-time. A defensive mistake allowed Danilo time and space down the right and, from the 23-year-old's excellent cross, Martinez got in front of his man and powerfully headed home.
Brahimi then darted through the midfield and drew a free-kick on the edge of the box. The former Rennes and Granada player took it himself, curling the ball over the wall and inside the post after 57 minutes.
It was 5-0 four minutes later, Lopez pouncing on a loose ball in the area and flicking it over the onrushing Chernik.
