Neymar was the provider for Suarez who slotted home into the corner after 60 minutes.
Suarez then crossed for Neymar to slide the ball home after 83 minutes.
The pair have been the driving force behind Barca in the absence of the injured Lionel Messi and now have 13 goals between them from five games.
"Obviously we miss Leo, he is the best in the world but I am going through a good run at the moment," Neymar told reporters.
"We are very pleased with the victory. For the third goal I made a long run for the ball and half the goal has to go to Suarez for the pass."
Barca lead Group E now with 10 points from three games while BATE have three points.
Fans turned up carrying Catalan flags and the UEFA anthem ahead of the kick-off was drowned out by whistles as they protested against a second fine for Barca, which they are appealing, for displaying political emblems.
Barca moved the ball around swiftly but the first real danger surprisingly came from BATE with keeper Marc-Andre ter Stegen beating away a fiercely struck free-kick by Igor Stasevich on the edge of the area after 17 minutes.
Andres Iniesta made his first start since picking up an injury at the end of September but Barca were dealt a blow with Ivan Rakitic limping off and he was replaced by Munir.
It was the youth product though that was hauled down by Filip Mladenovic after combining with Suarez, and Neymar dispatched the penalty.
Full-back Adriano clipped the outside of the post and Neymar headed just over before half time with Barca in control.
After plenty of pressure Barca finally scored again with Suarez showing once again his prowess in front of goal and then he found Neymar to wrap up the scoring.
(Reporting by Tim Hanlon in Barcelona, editing by Pritha Sarkar)
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