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. Finnish player Ville Leino, back at his former home stadium, then made it 2-0 on his debut for the Sabres.
Vanek then scored his second of the night in the same period. Pominville also finished with 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."
"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, however, hit the posts twice in the last period, through Francois Beauchemin and Luka Sbisa.