The hard-working winger had two goals in his first pre-season game last weekend and followed that with a goal and an assist as the Canadiens downed the Carolina Hurricanes 6-0 on Friday night.
"In the first game I was nervous at the start," said Bournival, the captain of the 2012 Memorial Cup champion Shawinigan Cataractes who spent last season with AHL Hamilton. "I'm getting more and more comfortable on the ice as it goes along.
"It's up to me to keep that going."
Greg Pateryn and Alex Galchenyuk scored in a 26-second span early in the first period, while David Desharnais, P.K. Subban and Max Pacioretty also scored for Montreal (1-1-1) before a disappointing crowd of 10,139 at the 15,00-seat Pepsi Colisee.
Peter Budaj, with 11 saves, and Dustin Tokarski, with 13, shared the shutout.
Cam Ward went all the way in goal for the Hurricanes (0-3-3), who will play a fourth game in as many nights when they face the Canadiens again on Saturday night at the Bell Centre. He faced 29 shots.
Bournival was named first star of the game, but a handful of rookies were impressive, including defenceman Magnus Nygren, who had two assists.
But coach Michel Therrien said it was veterans who led the way.
"The young players seized the opportunity to show what they can do," said Therrien. "But it was the veterans who set the tone of the game and that's positive.
"The Desharnais-Pacioretty line was very good at both ends of the ice. I also liked P.K. Subban with Josh Gorges, who played well together."
Desharnais, who comes from Laurier-Station near Quebec City, and Pacioretty each had two points.
"I always play well in this rink," said Desharnais, who visited often when he was in junior hockey with the Chicoutimi Sagueneens. "I have good memories here.
"I'm trying to be better each time out and I've noticed some improvement. I feel I'm quicker (than last season)."
Montreal was all over Carolina in the opening period and beat Ward four times on 10 shots.
Pateryn scored from the point on a Desharnais feed at 3:30 and Magnus Nygren's stretch pass gave Brandon Prust a breakaway that was stopped, but Galchenyuk followed in to score at 3:56.
Desharnais banked a shot in off Ward's pass from behind the net at 11:07 and Bournival, with his third in two pre-season games, fired in a Patrick Holland pass at 15:41.
Montreal had a two-man advantage when Subban, who also had two assists, blasted one in from the high slot in the second frame.
Pacioretty was awarded a penalty shot when he was tripped by Mike Komisarek and he beat Ward high to the stick side 4:22 into the third frame.
Therrien was in experiment mode, moving Galchenyuk to centre from left wing and using Swedish prospect Sebastian Collberg on right wing with Desharnais and Pacioretty. Everything worked against a Carolina team that was flat from the outset.