Andreas Dackell scored for the second time in as many nights and Radek Bonk and Magnus Arvedson added goals for the Senators, who are unbeaten in their last eight road games dating back to last season.
Jeremy Roenick scored for the Coyotes (0-1-0), who lost a home opener for the first time since moving from Winnipeg before the 1996-97 season.
The Senators, who beat Colorado 4-3 on Saturday, have posted a 2-1-4 mark in opening games since joining the NHL in 1992-93. But they failed in their only other chance to start 2-0-0, beating the Montreal Canadiens in the franchise's first-ever game only to go winless in their next 10 games.
Damian Rhodes, who made 22 saves for his second straight victory, got all the offensive support he would need in the first 24 minutes.
Phoenix goalie Nikolai Khabibulin blocked Sean McEachern's wrist shot from the low left slot, but couldn't stop Dackell's rebound from the right side as Ottawa took a 1-0 lead at 9:06 of the first period.
Bonk made it 2-0 when he rifled a shot from the right circle over Khabibulin's glove and inside the right post at 3:47 of the second period.
Gardiner added an unassisted goal just over three minutes later. Gardiner checked Jyrki Lumme off the puck against the end boards, stole the puck off Roenick's stick behind the goal and tucked a backhand shot into the net for a 3-0 Senators lead.
Arvedson extended Ottawa's lead to 4-0 when he scored from the low left slot 12 seconds into the third period.
The Coyotes spoiled Rhodes' shutout with 10:46 remaining when Dallas Drake stole the puck from Arvedson and passed to Roenick, who slipped a backhander into the net.
Khabibulin made 21 saves for the Coyotes, who have not beaten Ottawa since Dec. 13, 1996.
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