Buffalo erased a 2-0 deficit with goals in the second and third periods, but the Rangers prevailed to improve to 21-13-5 on the season, while the struggling Sabres fell to 10-20-9.
New York outshot Buffalo 42-33 in the game, and Rangers goalie Henrik Lundqvist was one of the contest's biggest standouts, making 31 saves.
In addition to his game-winning goal, Miller posted an assist, while defenseman Rasmus Ristolainen had one goal and one assist for the Sabres, and forward Kyle Okposo dished out two helpers.
Also, while Sabres goalie Robin Lehner allowed the OT winner, he enjoyed a strong outing, making 39 saves on 42 shots.
Lehner also embraced the nostalgic feel of the Winter Classic by donning an old-school mask during warm-ups, as seen in this video, courtesy of the NHL's official Twitter account:
For the Sabres, Monday marked their first participation in the Winter Classic since they hosted the inaugural event 10 years ago at Ralph Wilson Stadium in Buffalo, New York.
Things didn't get off to an ideal start for Buffalo on New Year's Day, as the Rangers needed just over four minutes to get on the board in the form of a goal from Paul Carey with assists from Boo Nieves and Jesper Fast, per the following video from NHL on NBC:
A little more than four minutes later, the Blueshirts struck once again when Michael Grabner lit the lamp for the 18th time on the year.
Grabner benefited from assists by Miller and forward Kevin Hayes, who set him up with a sweet spin move:
Buffalo entered the first intermission trailing by two, but it got back in the game just 56 seconds into the second period, when forward Sam Reinhart pushed a rebound past Lundqvist, as seen in this video from NBC Sports:
Ristolainen and Okposo recorded assists on the power-play marker, which came courtesy of a carryover penalty on Rangers defenseman Nick Holden near the end of the first period.
After USA Hockey announced its roster for the 2018 Winter Olympics in Pyeongchang, South Korea, during the second intermission, Buffalo struck again.
This time, it was Ristolainen, who beat Lundqvist top shelf on a goal assisted by Okposo and Ryan O'Reilly:
That goal came just 27 seconds into the third, and while there were plenty of chances for both sides, there was no additional scoring in regulation.
Three-on-three overtime made for a hugely entertaining extra session, and it got even more dynamic when Sabres forward Jacob Josefson was whistled for a tripping penalty.
New York took advantage, as Miller scored on a rebound off a shot from defenseman Kevin Shattenkirk to give the Rangers the win:
In their second Winter Classic appearance, the Rangers improved to 2-0, as they beat the Philadelphia Flyers 3-2 in 2012.
Buffalo, meanwhile, fell to 0-0-2 given it lost a shootout to the Pittsburgh Penguins in 2008.
The Rangers improved their playoff positioning too, as they are third in the Metropolitan Division with 47 points.
Buffalo earned a point in defeat. However, it remains last in the Eastern Conference with just 29 points.