Cody Hodgson, Tyler Myers and Henrik Tallinder scored for Buffalo, but the Sabres (2-13-1) continued their worst start in franchise history. The Sabres have lost their first nine home games, their longest stretch to open a season.
With the game tied 1-1 late in the second period, Anaheim scored two goals in 34 seconds. Vatanen took advantage of Matt Moulson's slow backcheck and fired a shot past goalie Jhonas Enroth from just inside the blue line. Moments later, Etem lifted a rebound opportunity past Enroth for his first of the night.
Thirteen players recorded points for Anaheim, which won its first game in Buffalo since 2009.
Buffalo barely tested Anaheim goalie Jonas Hiller, and the Sabres were outshot for the sixth straight game, 29-17.
The Sabres opened the scoring 7:15 in with a power-play goal. Matt Moulson fed Hodgson from behind the net for a low one-timed shot that beat Hiller to his right. Moulson and Hodgson had two points.
Anaheim tied it less than three minutes later when Getzlaf took a pass from Dustin Penner and lifted a shot over Enroth.
Perry's goals came in the third period, both wrist shots over Enroth's left shoulder.
It wasn't all good for the Ducks, whose power-play struggles continued. Anaheim has just one goal in its last 26 power plays after an 0-for-3 performance that included an abbreviated 5-on-3 late in the first period.
The three-goal loss was the Sabres' fifth of the season, four of which have come at home.
Mathieu Perreault had two assists for Anaheim.
The Sabres will head to the road for a three-game, West Coast trip that starts Tuesday in San Jose and concludes Friday against the Ducks.
NOTES: The Sabres placed RW Patrick Kaleta on waivers on Saturday after he finished his 10-game suspension for a high hit on Columbus defenceman Jack Johnson. ... The Ducks will finish their eight-game road trip on Monday against the New York Rangers. ... John Scott was among the Sabres scratches as he served the fifth game of a seven-game suspension for a late hit on Bruins forward Loui Eriksson.