Montreal stormed back from a 5-0 deficit early in the second with five unanswered goals to tie it at 5.
Ryder scored three of four straight even-strength goals by Montreal for his third career hat trick 6:52 into the third as the Canadiens closed to within 5-4.
Alex Kovalev, who scored Montreal's third goal 9 seconds before Ryder's third goal, rolled onto his back amid a wild ovation after he tied it with his team-leading 29th goal at 15:38.
Shanahan scored twice to reach 20 goals for the 19th straight season and Jagr had four assists to help stake the Rangers to their big lead.
Brandon Dubinsky and Sean Avery scored goals 14 seconds apart in the first, and Drury scored his 20th goal 28 seconds after Shanahan's second goal of the game to give New York a five-goal lead 5:03 into the second.
Shanahan scored New York's third goal of the first period and the first of two power-play goals 28 seconds apart by the Rangers early in the second to extend the streak that he began in 1988-89 with 22 goals in his second season with New Jersey.
Ryder drew Montreal within 5-1 with his 10th goal at 8:28 before cutting the lead to three with his second in a row at 12:52.
The Canadiens struck for two goals 9 seconds apart to cut the lead to 5-4 just 6:52 into the third.
Kovalev scored his 28th goal 6:43 into the third to make it 5-3. Ryder brought the crowd to its feet once again with his third goal of the game for his first hat trick since Apr. 7.
Dubinsky opened the scoring 11th goal 8:50 on a rink-length give-and-go play with Jagr off a Montreal turnover in the Rangers' zone.
Jagr also had a hand on the Rangers' next goal as Avery put away a rebound of Carey Price's pad stop on Jagr shortly after the faceoff that followed Dubinsky's goal.
Shanahan chased Price, who allowed three goals on 11 shots, with his 19th goal at 13:56. He got his second of the game against Cristobal Huet 4:35 into the second with Chris Higgins serving a double minor for high-sticking.
Shanahan, who also played for St. Louis, Hartford and Detroit over the course of the streak, surpassed Marcel Dionne last season for sole possession of the second-longest streak in NHL history.
He trails only Gordie Howe, who scored at least 20 goals for 22 straight seasons with Detroit from 1949-50 to 1970-71.
The game was delayed for a few minutes with 2:36 remaining in the third when a few hundred "Go Habs Go" signs given away to the sold-out crowd of 21,273 by a hardware retailer rained down onto the ice after Canadiens defenseman Mike Komisarek was called for his second minor penalty in a row.
