Nabokov stopped a short-handed shot by Stephen Gionta and a close-in attempt by Steve Bernier.
The Devils twice took the lead in their home opener and Grabner responded for the Islanders each time.
Grabner's second of the game came on a shot from the right circle that beat Brodeur on the short side at 13:43 of the middle period. Bailey had forced a turnover at center ice and made a pass that sent the speedy right wing into the Devils' zone.
"It was great plays by my linemates finding me," Grabner said. "I was trying to get to open ice and they made some great plays where I could use my speed and capitalize on my chances."
Ryder, who is wearing the No. 17 jersey that Ilya Kovalchuk had before returning to Russia after last season, rekindled memories of the Russian superstar by snapping a shot past Nabokov a little less than five minutes into the second period. It was his first goal with the Devils.
Brunner, who had a team-high eight shots in Pittsburgh, gave the Devils the lead 2:59 after the opening faceoff. Nabokov couldn't control a shot by Dainius Zubrus from the slot and Brunner tapped the loose puck into an open net for his first goal as a Devil.
Grabner, who was stopped on an early breakaway by Brodeur before the Brunner goal, didn't miss his second one-on-one. He took a pass from defenseman Travis Hamonic in stride and slid the puck under Brodeur to tie the game at 8:07.
NOTES: Josh Harris and David Blitzer, who head the group that bought the Devils this summer, dropped pucks at the ceremonial opening faceoff. ...This was the first of five regular-season games between teams. Islanders won last year's season series 3-2. ...Brock Nelson, the Islanders' first-round draft pick in 2010, made his NHL regular-season debut. He appeared in a playoff game last year. ... Other than Brodeur in goal, the Devils used the same lineup they did in Pittsburgh on Thursday.