Dany Heatley and Anton Volchenkov also scored for the Senators (2-1-1). Daniel Alfredsson returned to Ottawa's lineup one week after having knee surgery and assisted on two of the three first-period goals.
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 a bone chip from his right knee.
Shane Doan, Enver Lisin and Mikkel Boedker scored in the third for Phoenix (2-2-0).
Martin Gerber stopped 34 shots, taking 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.
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.
Notes: 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 ... A crowd of 20,179 watched the game, Ottawa's 38th straight sellout at Scotiabank Place ... The Senators are home again Saturday night to face the Boston Bruins (7 p.m., CBC, Team 1200).