Toffoli, acquired from Montreal a day earlier, scored, and the Flames won their season-high seventh straight game, beating the Columbus Blue Jackets 6-2 on Tuesday night.
"Obviously, coming to a really good team playing really good hockey right now, I was excited,'' he said.
On his goal at 11:04 of the third period. Toffoli chased down Johnny Gaudreau's lobbed pass and neatly put away his 10th goal of the season as he cut across the net and beat goaltender Elvis Merzlikins.
The Saddledome crowd of 9,639 immediately erupted.
"It was obviously pretty cool, but at the same time a little embarrassing for myself, (Rasmus Andersson) was all over me in the room,'' Toffoli said with a grin.
Eased into the lineup, getting only 12:18 in ice time, Toffoli managed four shots on goal.
"We were good right from the start,'' said Dube. "It was a lot of fun. It gave (Monahan) and I some energy to have Toff on the line and get going. I think we had to be good for him and I thought we built off a good start on the first shift.''
Erik Gudbranson, Elias Lindholm, Matthew Tkachuk, Dillon and Adam Ruzicka also scored for Calgary (27-13-6), which moved past Vegas into first place in the Pacific Division, one point ahead of the Golden Knights. The Flames also hold two games in hand.
Rookie goaltender Dan Vladar made his first start in 39 days and his first home appearance of the season for the Flames, and finished with 18 stops, improving to 7-3-1.
"Guys were just unreal in front of me,'' the Czech rookie said. "We are red hot right now. Everybody's playing really well, all the lines are just rolling and we're not just leaning on one line, every line can score.''
Merzlikins made his fifth straight start in goal for the Blue Jackets and faced 41 shots.
"They're a powerhouse team,'' Columbus coach Brad Larsen said. "They're big, they're strong, they've got a good balance of veteran guys. It's a pretty deep team. We can't have anybody not on board in that game, not one, to stay with them for 60 (minutes).''
First, Jack Roslovic's turnover resulted in a two-on-one for Gaudreau and Tkachuk. The Flames top scorers did not squander the chance, executing a perfect passing sequence culminating in Tkachuk's 22nd goal.
Then in the final minute, another turnover allowed Dube and Andrew Mangiapane to race in on a two-on-one, with Dube firing a shot past Merlizkins to snap a 13-game pointless streak.
Calgary opened the scoring two minutes into the game on the third goal of the season and second in as many games for Gudbranson. After Milan Lucic won a battle for the puck, sending a pass back to the point, with Gudbranson's low wrist shot squeaking through Merzlikins.
Making it 2-0 at 16:30, Lindholm centered the puck from the corner, and Blue Jackets defenseman Andrew Peeke, in attempting to clear the puck, instead swept it straight into his own net.
The Associated Press contributed to this report.