Tomas Plekanec drew Montreal even at 4-4 when he got credit for his second goal of the game at 11:02 of the third period.
Plekanec, who opened the scoring with a shorthanded goal at 2:04 of the first period, drove in on a breakaway and was stopped by goalie Craig Anderson, but the rebound went into the net off the skate of Ottawa defenseman Jared Cowen.
Bobby Ryan and Kyle Turris scored on the Senators' 18th and 19th shots of the second period against Price to give Ottawa its first lead at 4-3.
Erik Karlsson had a goal and two assists, and Clarke MacArthur had a goal and an assist for the Senators, who defeated Montreal in the first round of the 2013 Stanley Cup Playoffs. Anderson made 18 saves.
Price, who faced 38 shots in the first two periods, made a number of sensational saves in the second to protect Montreal's 3-2 lead before he was beaten on Ottawa's final two shots of the period.
The Montreal goalie, who will represent Canada at the 2014 Sochi Olympics, made a spectacular glove save on Milan Michalek's shot from the slot, the Senators' 30th of the game, at 9:06 of the second. Price stuck out his left pad to steal a sure goal from Ryan later in the period before making another glove grab to deny Mark Stone at 15:02.
Ryan broke through to tie it at 3-3 with his team-leading 19th goal at 17:16. Turris gave the Senators their first lead at 18:40 when he took a return pass from MacArthur and beat Price with a backhand for his 14th goal.
Plekanec, who will captain the Czech Republic team at the Olympics, fell to the ice after he was struck on the right leg by a shot by Ottawa's Cory Conacher midway through the first. Plekanec went to the dressing room briefly before returning.
Max Pacioretty made it 2-0 for Montreal with his team-leading 21st goal on a power play at 14:05.
David Desharnais, who missed the Canadiens' 4-1 loss to the New Jersey Devils on Tuesday because of the flu, scored at 15:36 to increase Montreal's lead to three.
Ottawa, which outshot the Canadiens 19-7 in each of the first two periods, scored two goals 1:12 apart late in the first to draw to within 3-2.
MacArthur scored his 17th goal at 17:13. Karlsson was left unmarked to score into a wide-open net at 18:25.