Allen made 13 power-play saves in his first shutout with the Canadiens and 22nd of his NHL career. His last shutout was March 8, 2020 for the St. Louis Blues.
It was the second-most saves in a shutout in Canadiens history. Jacques Plante made 52 saves in a 2-0 win at the Chicago Blackhawks on Nov. 13, 1955.
"[Allen] has been playing unbelievable lately," Canadiens forward Josh Anderson said. "We just haven't played good enough in front of him. He really deserved this one tonight. [He's] a good goalie, a really good goalie, so we know what we have back there and we're really confident in him each and every night."
Anderson had a goal and an assist, and Nick Suzuki had two assists for Montreal (2-6-0), which had not won at San Jose since Nov. 23, 1999.
"I thought it was a crucial game for our group," Allen said. "I know it's early, but we're obviously in a funk, and I thought it was an important game to get our momentum going so we can finish this trip on the right note."