Weber found the back of the net 2:12 into the extra frame to give Boston its second Clarkson Cup in three years.
"I don't know, it just happened so fast and I haven't yet realized what happened," she said.
"Last year in college I scored in overtime to win the Mayor's Cup between Brown and Providence in our city, so we won last year in overtime and that's pretty big," Weber added. "(But) now that I think about it, it is the biggest goal I ever scored."
Hilary Knight and Brianna Decker had a power-play goal and an assist apiece for Boston in regulation. Ann-Sophie Bettez and Emmanuelle Blais scored for Montreal with Caroline Ouellette chipping in with two assists.
Boston and Montreal played for the league's top prize in 2013, when the Blades also triumphed over the Stars. The seventh CWHL championship marked the Blades' second title and third straight appearance in the final.
"Winning this is one of the biggest things you can do in women's hockey and I'm so proud of our team to go through everything we did and win it in overtime," said Weber, an Austrian native who will join her national squad next month at the IIHF Division I world championships in Poland.
Boston advanced to the Clarkson Cup after topping last year's champs, the Toronto Furies in a best-of-three playoff series.
Montreal jumped into the championship game after back-to-back shutouts from Charline Labonte against the Calgary Inferno. Labonte, who played 136:29 without allowing a goal, earned tournament MVP honours.
"Labonte played a hell of a game, hell of a tournament so it was good to get one in there," said Decker, who led all scorers in the Clarkson Cup with five goals and three assists, including three on the power play and the only short-handed goal.
Montreal scored 31 seconds into the game when Bettez found the puck after a bounce off the boards. Ouellette and Blais assisted.
But Knight tied the game on the power play 16 minutes later.
"We just needed to get one in there," said Knight.
After a scoreless second period, Decker gave Boston the lead at 6:17 of the third off passes from Monique Lamoureux and Knight.
Just over three minutes later, Blais scored for Montreal on the Stars' first shot of the period. Blades goaltender Genevieve Lacasse argued there was a distinct kicking motion, but officials disagreed and the score stood at 2-2.
Boston dominated puck possession and had 28 shots on goal to Montreal's 14.