Miller won it with a one-timer from the right circle after Sabres forward Marcus Johansson tied it 5-5 with 59 seconds remaining in the third period on a one-timer from the point with goalie Carter Hutton pulled for an extra skater.
Buffalo tied the game three times, including twice after trailing by two goals.
WATCH: [All Sabres vs. Canucks Highlights]
"I don't think we blew a lead, that's not really the word I would use," Miller said. "It's not like they were dominating. They just got a couple of looks on breakdowns and scored. We weren't pouting."
Sabres center Jack Eichel appeared to win it at 2:09 of overtime, but teammate Henri Jokiharju was called for an interference penalty.
"It takes a goal out of their net and puts one in ours and that's frustrating," said Eichel, who had two assists to extend his point streak to 12 games (10 goals, 12 assists), his NHL career high and longest active streak in the League.
Antoine Roussel and Josh Leivo each scored two goals, and Tyler Myers scored his first goal with Vancouver (15-11-4), which has won three of its past four. Thatcher Demko made 23 saves.
"We played a pretty good 5-on-5 game," Leivo said. "That's a good team with a lot of skill. They capitalized on their opportunities. We're just happy to sneak away with two points."
Victor Olofsson and Sam Reinhart each had a goal and an assist for Buffalo (13-11-6), which has lost four of its past six (2-1-3). Hutton made 24 saves and is 0-5-4 in his past nine starts.
"We get a point out of it, but I don't know how to explain right now, it's tough," Hutton said. "It feels like I'm swimming out there a bit."
Hutton started the season 6-0-0 but hasn't won since Oct. 22.
"Honestly, it's tough to put into words right now," Hutton said. "It wears on me for sure."
Zemgus Girgensons got Buffalo within 5-4 with 7:55 remaining in the third.
