The 33-year-old defenseman missed 73 games following surgeries on his right knee and left foot. He had an assist in a losing effort in his first game back on Thursday.
''The biggest thing is winning,'' said Weber, who gave the Canadiens a 2-0 lead after one period. ''If you help out by scoring goals, or stopping goals, whichever way it is, it's coming in here after getting two points that I definitely miss the most.''
The Canadiens outshot New York 42-22 and beat the Rangers for the fourth straight game at the Bell Centre.
Montreal went up 2-0 in the first period on Weber's first two goals of the season. He was playing just his second game since returning from an injury.
Weber scored 3:53 into the game, sending a one-timer from the blue line past Georgiev. It was Weber's first goal since Dec. 5, 2017.
''The guys on the bench were all so happy when he scored his first goal,'' said Canadiens head coach Claude Julien. ''He's earned that respect from his teammates, from us, just by the way he handles himself. Everyone is extremely happy for him tonight. Second game, scores a couple of goals, sets the tone and has a real solid outing.''
He netted his second of the game on a great play by Tatar. Lying on the ice with his back to the play after being tripped at the blue line, Tatar spotted a streaking Weber and fed him with a backhand pass for the partial break. Weber went high to the glove side and scored with eight seconds remaining in the first period.
''That was a crusher,'' Rangers coach David Quinn said. ''To give that goal up in the fashion that we did really was a tough pill to swallow and tough to overcome. You know we're not in a great mindset right now, and to come in the second period down 2-0 playing the way we did the last seven or eight minutes and give up a goal of that fashion really set us back.''
Lehkonen doubled Montreal's lead - and his goal tally for the season - with two quick scores early in the second.
The Canadiens went up 3-0 when Lehkonen beat Georgiev with a wrist shot on a 2-on-1 at 2:04. The Rangers goalie smashed his stick against the post in anger after the goal.
Lehkonen made it 4-0 at 5:29, a quick release on Georgiev's blocker side after a giveaway by Neal Pionk behind his own net.
With the teams playing 4-on-4, the Rangers got one back on a wrist shot by Vesey at 8:10 of the second.
Strome, on the power play, made it 4-2 with 5:49 remaining in the middle frame.
Tatar, with his 11th of the season, restored Montreal's three-goal lead with a power-play goal at 10:28 of the third.
NOTES: The Canadiens placed former first-round draft pick Nikita Scherbak on waivers earlier on Saturday. ... Speedy forward Paul Byron was back in the lineup for Montreal after missing 14 games with a lower-body injury.
Montreal: Hosts the San Jose Sharks on Sunday.
New York: Hosts Winnipeg on Sunday.
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