Vanek, the top goal-scorer for Buffalo the last five seasons, set the tone with his first goal at 4:02 in the first period. Ville Leino of Finland, playing in his former home stadium, then made it 2-0 on his debut for the Sabres.
Nate Guenin replied for Anaheim in the second period, but Vanek helped restore the Sabres' two-goal lead by setting up Jason Pominville.
Vanek then scored his second of the night later in the period. Pominville also had two assists, as did Luke Adam.
''The puck was bouncing for us a bit tonight and we played a good first period with the two-goal lead,'' Vanek said. ''We tried to get pucks to the net and it paid off early on.''
Buffalo coach Lindy Ruff was happy with the way Leino played on a line with Brad Boyes and Tyler Ennis.
''Leino did a heck of a play, taking the puck from his own skates before scoring,'' Ruff said.
Doing so in his first game for a new club was nothing new for Leino, who dominated the ice for Finnish team Jokerit in the same arena before leaving for the NHL.
''I have done it for every team where I've played, and was lucky to do it again,'' Leino said. ''It was cool to do it here in Finland.''
Anaheim coach Randy Carlyle felt his team had allowed Buffalo an easy game.
''We did not do a very good job of transferring the puck from our defense to the forwards, and had to spend the first half of the game too much in our own end,'' he said.
''We played more as individuals in parts of the game versus a team.''
Anaheim hit the post twice in the last period on shots by Francois Beauchemin and Luka Sbisa.