The 24-year-old centerman extended his scoring streak to five games.
For his part, Suzuki collected his third assist of the contest on Domi's marker.
The Panthers didn't take long to respond, though. Just 1:29 later, Barkov's second goal of the game made it 3-3.
Jonathan Huberdeau put Florida up 4-3 with 6:30 left before the break.
Thirty-five seconds later, Huberdeau struck again and put head coach Joel Quenneville's contingent up by two.
The Canadiens were down 5-3 through two periods.
Panthers defenseman Mike Matheson further extended Florida's lead just 45 seconds into the third period.
But a power play goal by Jesperi Kotkaniemi cut into the Panthers' lead again.