Higgins scored a short-handed goal 7:22 in before extending the Canadiens’ lead to 2-0 with his second of the game 14:11 into the second.
The Smithtown, N.Y., native completed his first career hat trick when he scored Montreal’s fourth goal on a breakaway 16:35 into the third.
Guillaume Latendresse also scored for Montreal, which had lost consecutive games (0-1-1) for the first time this season.
Price stopped 28 shots, including a right pad save on Dany Heatley’s breakaway shot 7:36 into the third, for his fourth career shutout.
Alex Auld made 27 saves for Ottawa, which lost its second straight after going unbeaten in regulation over its previous five games (4-0-1).
Higgins opened the scoring when he beat Auld with a shot from the left side with 3 seconds remaining in defenseman Ryan O’Byrne’s delay of game penalty for putting the puck over the glass.
Robert Lang got the Canadiens’ second delay of game penalty for the same infraction 11:52 into the second, but Montreal’s penalty-killing unit was so successful that at one point it appeared as if it was conducting a 4-on-5 power play.
Higgins finished off a fine pass by Sergei Kostitsyn with a shot from the right side past Auld at 14:11.
Lang had a hand in the Canadiens’ third goal at 15:39 when he won a draw against Ottawa’s Dean McAmmond and drew the puck back in the Senators’ zone.
Latendresse stepped into the bouncing puck and drove a slap shot over Auld’s left arm into the top right corner for his first goal in 11 games.
Auld has started eight of the Senators’ last nine games. … Price’s last shutout also was against Ottawa, a 3-0 win on April 1 for the third shutout of his rookie season.