Shane Doan, Joel Perrault and Derek Morris also scored for Phoenix, which tied a club record for goals on opening night. The Nagy-Doan-Reinprecht line had four goals and seven assists. Viktor Kozlov scored twice for New York. Miroslav Satan also scored for the Islanders.
Phoenix blitzed the Isles after New York jumped ahead 1-0 at 1:45 of the first period on Satan's shot from the top of the right circle.
After New York's Tom Poti was sent off for tripping, Phoenix tied it at 5:25 on Nagy's power-play goal. Nagy was originally credited with an assist after he slipped the puck to Reinprecht in the crease, but the official scorer gave Nagy the goal five minutes later.
It was Nagy's first goal since Jan. 14. He missed the final 29 games of last season with a knee injury.
The Coyotes made it 2-1 on Morris' slap shot from the left circle at 9:35 of the first.
Phoenix finished the first period with 15 shots on goal - an encouraging sign for a team that often struggled to muster offence last season.
Phoenix put the game away with three goals in a 2:19 span of the second period.
Nagy's second goal came at 7:34 after he slipped behind the defence and took a pass from Doan. Nagy's shot was headed wide of the net but bounced in off the skate of Islanders defenceman Brendan Witt.
At 9:02 Doan took a nifty backhand pass from Reinprecht in the crease and beat Rick DiPietro to his glove side to put Phoenix ahead 4-1.
Fifty-one seconds later, Perrault scored his first NHL goal on a breakaway.
Reinprectht gave Phoenix a 6-1 lead when he redirected Ed Jovanovski's blast from the blue line on a power play at 12:11 of the second period.
DiPietro allowed six goals in 25 shots before being replaced by Mike Dunham.
The Islanders' second goal came when Kozlov scored on a rebound of a shot by Witt at 3:04 of the third period. Kozlov scored again at 6:07 to cut the deficit to 6-3.
Phoenix goalie Curtis Joseph earned his 429th career victory. That leaves him eight shy of Hall of Famer Jacques Plante, who ranks fifth in NHL history.
Notes: The Islanders opened on the road for the 30th time in 34 years. After stops in San Jose, Los Angeles and Anaheim, the Islanders open at home against Boston on Oct. 14. ... The victory evened Phoenix coach Wayne Gretzky's career record at 39-39-5.