Radek Dvorak scored a deflected goal in the first period to help the Hurricanes win their second straight following a five-game slide that coincided with a lower-body injury to starting goalie Cam Ward.
Peters, making his sixth straight start, made one of his best saves in the final minute of the second period when he stopped a streaking Michael Grabner after a turnover.
Peters finished with the Hurricanes' first shutout of the season and his first since blanking Washington last March 12.
All six of Carolina's wins have come by one goal.
Backup Kevin Poulin made 23 saves in his second start in six days for the Islanders. They have lost two straight and four of six, and were shut out for the first time this season.