Hammond improved to 8-0-1 in nine starts for Ottawa (31-24-11), which remained seven points behind Boston for the second wild card into the Stanley Cup Playoffs from the Eastern Conference.
"We know that to make the playoffs we need to go on a run here and to not have anyone pack it in and keep fighting when we're down 2-0 just shows the care in the locker room and that we're going to see this thing through, and I think it's just a big asset that we have that character in here," said Hammond, who has allowed two goals or fewer in each of his first nine NHL starts.
The 27-year-old is the fourth goaltender in League history to accomplish that feat; he's the first since Frank Brimsek, who took over from fellow Hall of Famer Tiny Thompson with the Boston Bruins early in the 1938-39 season, did it in his first 12 starts. Brimsek went on to win the Calder and Vezina trophies, was named to the First All-Star Team and led the Bruins to the Stanley Cup.
"I don't want to take anything away from his game [Thursday], but it was one of his easiest games because I think it was the best we've played in front of him," Ottawa coach Dave Cameron said. "When your team is down 2-0 early, through no fault of his, you look for your team to rebound, but more importantly you look for your goalie to rebound, and he held the fort. He held the fort and we played much better, and from top to bottom I thought we were real solid the last two periods."
Hammond is the third goalie in the past 25 years to earn at least one point in his first nine NHL starts. Patrick Lalime accomplished the feat with the Pittsburgh Penguins in 1996-97, and Ray Emery matched it with Ottawa from 2002-06.
"Again, a solid game, made some key saves when he had to and made the easy saves look really easy," Karlsson said. "He plays the puck well and is really solid for us and easy to play with him behind us."
Karlsson scored at 2:50 of the second to get Ottawa within 2-1, assisted on Mark Stone's go-ahead power-play goal 1:23 into the third period and scored his second of the night and 18th of the season at 9:43 to make it 4-2. That goal came one second after the Senators killed Matt Puempel's double minor for high sticking.
Max Pacioretty scored two special-teams goals 2:11 apart in the first period for Montreal (42-19-7), which led 2-0 before Ottawa had its first shot on goal.
"I think we came out and didn't really have the jump we wanted," Karlsson said. "But we stuck to it, and from the second period on we just kept playing hard and we stayed on pucks and we got it in deep and scored on the opportunities that we were given, and controlled the game from there."
"Did they get some fortunate bounces? They did, but they worked hard for their bounces," Canadiens defenseman P.K. Subban said. "I find that when you're working hard you get those types of bounces. They worked extremely hard tonight. We knew that they're a team that's battling to make the playoffs, they're battling for their lives every night, and we had to match that intensity."
Montreal leads the Atlantic Division with 91 points; the Tampa Bay Lightning also have 91 but have played one more game. The Canadiens are second in the Eastern Conference to the New York Rangers, who also have 91 points but have played two fewer games than Montreal.
"We're not catching anybody by surprise anymore," Price said. "When we're sitting at the top of the standings other teams are prepared."
Karlsson beat Price with a slap shot to begin Ottawa's comeback. Condra tied it at 18:21 of the middle period when he got credit for his ninth goal of the season after Jean-Gabriel Pageau's pass from behind the net went into the net off his leg.
Karlsson got the second assist when Stone deflected Patrick Wiercioch's shot past Price for his 17th goal, putting the Senators ahead to stay.
Karlsson's second goal came off a pass from Milan Michalek. Methot completed the scoring at 12:36 with his first goal since March 10, 2014.
Pacioretty made it 1-0 at 6:43 of the first period with his 32nd goal. He deflected Subban's power-play slap shot from the point past Hammond with 14 seconds remaining in Alex Chiasson's double minor for high-sticking Montreal center Tomas Plekanec.
Pacioretty scored his 33rd of the season at 8:54 when he deked on a breakaway and put a backhand past Hammond for his first shorthanded goal of the season.
Montreal has gone 146:15 without scoring an even-strength goal.
"I think we had some opportunities to put the game away," Subban said. "When you're up 2-0 you want to get that third goal to really put the dagger in, but we didn't get it."
Senators defenseman Mark Borowiecki left the game in the third period after Subban's shot deflected off his stick and hit him in the face.
"It was pretty hard," said Borowiecki, who had five stitches on his left cheek. "Those ones that come up your shaft they seem to pick up speed and hits you pretty good, so I was a little scared at first but it feels good."
Canadiens defenseman Sergei Gonchar played his 1,300th game. He is the 57th NHL player to reach that milestone.