David Moyes recalled Yakubu to partner Andrew Johnson up front for the Toffees, while the home side were captained by former Leeds United midfielder Eirik Bakke.
Everton enjoyed a majority of the early possession, but Brann threatened after nine minutes as Azar Karadas headed over from close range from Jan Gunnar Solli's free-kick.
Phil Jagielka then did well twice to deny Thorstein Helstad a shooting chance as the Norwegian champions began to get the upper hand.
Twenty-five minutes in Osman fired a 20-yard shot straight at home keeper Hakan Opdal, who was extended eight minutes later by the same player, following a good interchange of passes between Yakubu and Johnson.
Right on half-time Brann thought they had the lead as Helstad's corner from the left was knocked into his own net by Tim Howard, but the referee had already blown for a foul on the keeper by Bakke.
Eight minutes into the second half Tim Cahill's superb 25-yard left-foot shot was tipped on to the post by Opdal, but the home keeper was beaten six minutes later as Everton took the lead.
Johnson turned away from his marker down the right and when his cross was headed out to the edge of the area, Osman flashed a low, first-time right-foot shot into the far corner of the net.
Manuel Fernandes forced Opdal into a save with a drive from distance 13 minutes from time, before Brann threatened an equaliser twice in the space of a minute.
First Howard had to push away a 25-yard effort from debutant Njogu Demba-Nyren and from the resulting corner, the American had to get down low to gather Helsatd's header right on the line.
However, with two minutes left on the clock, Everton doubled their advantage as Joleon Lescott powered forward from left-back to deliver a perfect far-post cross which was tucked away by substitute Anichebe.