Jaroslav Spacek had a goal and assist, and Clarke MacArthur also scored for the unbeaten Sabres, who won their fourth straight to start the season.
It's the third time in franchise history Buffalo has opened with four wins, and second time in the last three years.
"Well, it's not easy," said Vanek, who has scored in each of the Sabres' first four games this season. "We got an early jump on them, and from there on out we played pretty well."
That they did. After busting out with a three-goal first-period that included two power-play goals, the Sabres cruised the rest of the way -- thanks to a stingy defense that has allowed just five goals, and the play of the four-year veteran Vanek.
"I think he has more confidence," Kotalik said. "He's cleaned up some little stuff, and now he's a real fine player."
Kotalik was the beneficiary of Vanek's hard work in the third period when Vanek outraced Vancouver defenseman Shane O'Brien to the puck along the left boards at the Canucks' blueline before cutting to the net. O'Brien hauled Vanek down from behind, but Vanek was still able to control the puck and serve a perfect pass that Kotalik one-timed past Vancouver goalie Curtis Sanford from the high slot to give Buffalo a 5-2 lead.
"I'm looking at the effort behind the plays," Sabres coach Lindy Ruff said. "I'm looking at the play made on that last goal. It's that effort that really sticks out."
Ryan Miller finished with 23 saves for the Sabres.
Sanford had 26 saves for Vancouver in his first start of the season in place of Roberto Luongo. The Canucks' captain got the night off after allowing 12 goals in his last three starts.
Buffalo put this one away early, scoring three straight goals in a 12-minute span in the opening period. Spacek started the blitz with a power-play goal just a minute in, and MacArthur made it 2-0 with Buffalo up a man at 8:30.
Kotalik increased the advantage to 3-0 with just under 7 minutes to go in the first when his wrist shot from the high slot squirted through Sanford's pads.
"I wasn't playing my game in the first period and it showed," Sanford said. "I was fighting the puck a little bit, and all of a sudden the bounces were going their way."
Kesler closed the gap with his third of the year just 2 minutes into the second period, but Vanek lifted a backhand over Sanford's shoulder from in tight midway through the second to restore Buffalo's three-goal lead.
He didn't get his sixth goal last season until the 21st game.
"I felt pretty good during the second half last season, but I'm showing up on the scoresheet now and I think people are making a big deal out of it," Vanek said. "I've been on hot streaks before and then got pretty cold, so now I'm just trying to be consistent."
Bernier made it 4-2 early in the third period before Kotalik capped the scoring with 5:49 left.
"Anytime you spot a team that's only given up three goals in three games a three-goal lead right off the bat, you're obviously not giving yourself a chance to win," Vancouver coach Alain Vigneault said.
Both teams entered the league in the 1970 expansion class. ... This was the first meeting between the clubs since Buffalo won 4-3 in a shootout in January 2007. ... The Sabres killed off all three Vancouver power plays to go 18-for-18 on the penalty kill this season. They came in as one of only four teams that have not allowed a power-play goal. ... Buffalo had not surrendered a second-period goal this year until Kesler's tally. ... Bernier was acquired in a trade with the Sabres this summer. ... Eight of Spacek's last 10 goals dating back to last season have come on the power play.