Cole scored three goals in the first 5:41 to set a Montreal record for the fastest hat trick at the start of a game, and the Canadiens beat the Senators 5-1 on Friday night.
"Something was mentioned to me (about the record), but it's not really in the forefront of my mind right now," said Cole, who matched a career high of 30 goals for the season.
Patteri Nokelainen also scored in the first period, and Lars Eller added one in the third for Montreal, 4-0-2 against Ottawa.
"I'm glad we don't play them anymore," Senators coach Paul MacLean. "They certainly bring the worst out in us and tonight we weren't as prepared to play the game and we didn't play with enough discipline."
Jason Spezza, playing his 600th NHL game, scored for Ottawa to reach 30 goals for the fourth time in his career and the first time since 2008-09.
The Senators, 1-3-2 in their past six games, are seventh in the Eastern Conference, two points ahead of Washington and Buffalo
"Pressure's a wonderful thing," MacLean said. "You can find out a lot about yourself and about your team, and how hard it is in the league. For some reason we feel we have that on us and we have to find a way to handle it or channel it into the right spots."
Cole scored on the first shot on Anderson, who had a routine shot go off his glove and into the net only 29 seconds in. Cole then scored off P.K. Subban's rebound to chase Anderson.
Cole completed the hat track at 5:41 against Bishop.
Montreal defenseman Yannick Weber left after the first period because of a lower-body injury. ... Bishop had made the last seven straight starts for Ottawa. ... Spezza has 608 career points.