New Jersey's Damien Brunner produced the first and last goals of regulation, the second of his two scores creating a 3-3 tie with 12:26 left in regulation.
Brunner was the first player to make an attempt in the shootout and he, like the next 10 to take the ice, missed his try.
Moulson finally brought an end to the marathon shootout by getting the best of Martin Brodeur.
Brodeur started New Jersey's home opener after sitting out the Devils' first game of the season on Thursday against Pittsburgh. He had started New Jersey's season opener for an NHL record 18 consecutive years.
Evgeni Nabokov made 26 saves before stopping all six New Jersey attempts in the shootout. Brodeur turned aside 23 shots.