Alfredsson scored on Ottawa's third shootout attempt and the Senators extended their winning streak to four with a 3-2 win over the Montreal Canadiens on Saturday night.
Anderson, who got his first shutout of the season Thursday night at the New York Rangers, was perfect after making 33 saves in regulation and overtime.
"Andy was huge in the shootout by stopping all three shots," said Alfredsson, who has scored on each of his four shootout attempts this season. "They played well. Andy played unbelieveable and he gave me a chance to win it for us."
Kyle Turris and Jason Spezza scored in the third to give Ottawa a 2-1 lead.
Max Pacioretty scored with 38.9 seconds to tie it at 2 with Carey Price pulled for an extra attacker.
"You don't want to give up a lead in the last minute," Anderson said. "They had a couple of whacks at it. It's frustrating but at the end of the day we found a way to get the win."
The Senators face Winnipeg on Monday night before going back on the road for their second six-game trip of the season.
"We found a way to get the two points and that's the result we wanted," said Spezza, who scored his 19th goal on a power play 13:42 into the third to put Ottawa up by one after Tomas Plekanec was given a double minor for high-sticking Zack Smith in the face.
Plekanec scored a short-handed goal on a breakaway 42 seconds into the third to put Montreal up 1-0.
Price made 23 saves for the Canadiens, including a sprawling stick save to deny Bobby Butler's scoring chance moments after Plekanec's goal.
Montreal, which lost its third in a row, fell to 1-6 in shootouts this season.
"We have to find a way to win those," Price said. "I don't really know what to say. We've just got to find a way to get the extra point."
The Canadiens have won only three times in their last 13 games, including a 3-8-1 mark since interim head coach Randy Cunneyworth took over from Jacques Martin, who was fired on Dec. 17.
Montreal had an apparent goal by P.K. Subban disallowed 7:12 in. Referee Paul Devorski waved off the tally and called a goaltender interference penalty on Andrei Kostitsyn, who appeared to have been pushed into the crease by Senators left wing Milan Michalek.
"To me it was a legitimate goal that should have counted," Cunneyworth said.
Bourque, who is eligible to make his Montreal debut on Sunday night against the New York Rangers, was acquired from Calgary on Thursday in a multiplayer deal that sent Michael Cammalleri to the Flames.
Scott Gomez returned to the Canadiens' lineup after missing 21 games because of a groin injury. A center throughout his 12-year NHL career, Gomez played left wing on center Tomas Plekanec's line.
- Senators D Brian Lee was left out of the lineup.
- Gomez, who missed nine games earlier in the season, had four assists in 14 games.
- Canadiens D Alexei Emelin did not dress for the game.