The UEFA Cup last 32 first-leg victory in Norway was Everton's sixth successive European victory, a total no previous Goodison Park manager has achieved.
And teenage striker Victor Anichebe, with his fourth in the competition this season, is one short of setting a club record for a European run.
Leon Osman scored the first goal before Anichebe sealed the win with two minutes remaining.
"Brann are a good side. They have just won the Norwegian championship and beaten a top side like Rosenborg to do it, so we know we have produced a very good victory here.
"But we needed to improve after the break. We seemed to be trying to do everything perfectly, it was causing us problems and we were losing possession.
"But we got better as the game went on and Leon Osman and Victor scored fine goals.
"I will not start thinking about the next round until we have completed the job, the second leg being next week.
"We need to show respect because Brann showed themselves to be a very good side and we know they will come to Goodison Park desperate to get revenge for this.
"But we are getting our players back from injury and the African Nations Cup. Mikel Arteta should be back from injury by the time the second leg comes around and we have players suited to European competition becoming available again.
He added: "I was also pleased with the performance of (Ayegbeni) Yakubu, he came back into the side and played really well up front. He kept hold of the ball, linked the play well and worked really hard for the team."