It was Byron's fourth goal of the season, and the lone assist went to Joel Armia.
Julien's squad took that lead into the final frame.
Brock Nelson finally got the Islanders on the board at the 6:14 mark of the third period.
He was the beneficiary of a Carey Price mistake behind the Canadiens' net, which resulted in a loose puck coming his way off the stick of Josh Bailey.
Jordan Weal added another goal for the Canadiens to give Montreal a 5-1 lead with 4:32 remaining in regulation.
It was his eighth goal of the season, matching his career-high.
Ryan Pulock responded for the Islanders with a power-play goal, but Armia added an empty-netter to cap the scoring.
It was Armia's 16th goal of the season.
That's the way things finished up in Brooklyn.