Dany Heatley and Anton Volchenkov also scored for the Senators. Alfredsson returned to Ottawa's lineup one week after having a bone chip removed from his right knee and assisted on two of the three first-period goals.
"He definitely has a different body than everybody else -- I've never seen a guy healing as quick as he does," said goalie Martin Gerber, who stopped 34 shots.
Alfredsson set up goals by linemates Spezza and Heatley 3:08 apart late in the first to put Ottawa up 3-0. The Senators' captain missed one game -- Saturday's 3-2 loss to Detroit -- after having arthroscopic surgery one week earlier to remove the bone chip that had become dislodged during the team's season-opening, two-game series in his native Sweden.
"I feel better now, I guess, than I did before even because I had a little bit of grinding in the knee that I don't feel now, so that's obviously a positive," said Alfredsson, who added that he would be in the lineup Saturday against Boston.
Alfredsson, Spezza and Heatley combined for seven points.
Olli Jokinen, who had two assists to increase his total to six points in four games, was not surprised by Alfredsson's quick return.
"There's a reason why he's the captain, why he's an elite player in this league," said Jokinen, who was Florida's captain before he was traded to the Coyotes during the June entry draft. "He carries a lot of the load on that team and he leads by example."
Gerber took a shutout bid into the third period when Doan scored his third goal in four games at 2:02 to draw Phoenix to 4-1.
Ilya Bryzgalov made 20 saves for the Coyotes.
"We're going up against some good teams," Phoenix coach Wayne Gretzky said. "We need some of our guys to answer the bell a little bit. I know Bryzgalov's better than he played tonight, and he didn't step up. We need some more people to step up."
Jokinen said that the way the Coyotes played in the first 40 minutes was "embarassing."
"I think the third period was the way we can play," Jokinen said. "We have a young team over here and we've just got to find a way to put a solid 60 minutes together. That's what we were doing in the first two games. We have a lot of speed in this lineup so we have to use that a little more."
Coyotes left wing Daniel Carcillo sat out as he began serving a two-game suspension handed down by the NHL earlier in the day. Carcillo, who will also sit out Saturday's game in Montreal, was suspended without pay for coming off the bench Wednesday to fight Chicago's Ben Eager.
Ruutu, who signed with Ottawa as a free agent on July 2, drove the left side and put a shot over Bryzgalov into the top left corner of the net 8:31 in for his first goal with the Senators.
Alfredsson set up Spezza at 15:10 for his 850th career point, before adding his 851st at 18:18 on Heatley's fourth goal of the season.
Spezza got his third straight point when he combined with Nick Foligno to set up Volchenkov's first goal 7:05 into the second which put Ottawa up 4-0.
Ruutu's second score put the Senators up 5-1 at 4:09 of the third period.
Lisin, who made his season debut, cut Ottawa's lead to three again when he put a one-timer past Gerber from short range at 9:00. Boedker, Phoenix's 18-year-old first-round pick in June, drew the Coyotes within 5-3 with his second goal at 14:39.
Spezza scored his third goal of the season, his second of the game, on a power play with 1:43 left.
"There were two different games, but you know what, we hadn't played in a week so I'm happy the way things went when we were playing well," Senators coach Craig Hartsburg said.
- Coyotes RW Brian McGrattan and D Ryan Lannon did not dress. McGrattan, acquired from Ottawa on June 25, has yet to play for Phoenix.
- Senators D Luke Richardson and LW Christoph Schubert were healthy scratches.