Trailing 2-0 heading into the third period, Vancouver rallied with a pair of goals, and goaltender Thatcher Demko forced extra time with solid saves in the final minutes.
Demko believes the win will help galvanize the struggline team.
"It's a test of the will and the camaraderie in the room for sure," Demko said.
"Pucks haven't been going in for us very easily last couple games, but I thought we played really hard tonight, regardless, and it's easy to kind of stray away from that when things aren't going your way, but it was actually quite the opposite in the third there. Guys are picking each other up and working hard for each other and it ended up paying off."
Demko stopped 20 of 22 shots. Igor Shesterkin made 33 saves for the Rangers.
Miller and Vasily Podkolzin scored in regulation for the Canucks (4-5-1), who won at home for the first time this season and snapped a three-game losing streak. Conor Garland contributed a pair of assists.
The Rangers (6-2-2) got goals from Mika Zibanejad and Artemi Panarin. Adam Fox had two assists.
Oliver Ekman-Larsson's hooking penalty gave the Rangers 36 seconds of 5-on-3 with less than four minutes to play in regulation.
The Canucks killed off the two-man advantage with strong play Demko, weathering the second penalty.
Vancouver's netminder lost both his stick and blocker and still managed to make several kick saves - with his bare right hand behind his back - to keep the score tied at 2-2.
Miller and defenseman Tucker Poolman dove into the crease to help out their goaltender during that scramble.
"I'm just glad those pucks didn't hit me and they hit him," Miller said. "I'm just there for a little second line of defense. I saw he didn't have any of his gear on for a second there.
"It's a full scramble and panic. That was nuts. But you're just trying to keep the puck out of the net at all costs."
Canucks head coach Travis Green said killing off that 5-on-3 was unlike anything he'd ever seen.
"I think it just showed the desire of our group to sell out and find a way to get the job done and we had (Demko) making a couple unbelievable sales, guys laying their bodies on the line," he said.
"That's what you want to see out of your team,. A heavy desire to win, a will to win that takes you to levels that maybe you don't always do."
At the other end, Shesterkin stymied the Canucks for two periods before conceding a goal 1:29 into the third.
Garland looked ready to shoot near the goal line but instead sliced a pass to Miller, who tapped a shot in back door for his second goal of the season.
Minutes later, Garland dug the puck out from behind the Rangers net and flipped it to Podkolzin. The Russian rookie fired a rocket from the face-off dot to even the score.
Vancouver's offense began clicking late in the second period with Elias Pettersson and Miller streaking down the ice for a 2-on-1 chance. Pettersson ripped a shot from the bottom of the faceoff circle, only to see the puck hit the crossbar.
Moments later, Podkolzin muscled through New York's defense and tried unsuccessfully to get a backhanded shot past Shesterkin.
The Rangers led 2-0 midway through the second period with their second power-play goal moments after killing off their fifth penalty.
Ryan Lindgren had just stepped out of the box when the Canucks were called for too many men and Panarin was quick to capitalize, unleashing a shot from near the blue line that hit Vancouver defenseman Tucker Poolman and sailed past Demko.
Zibanejad opened the scoring with a power-play score 2:33 into the second after Canucks defenseman Tyler Myers was booked for interference.
Panarin threw a puck into the slot and Zibanejad redirected it up and over the sprawling Demko. It was the Swedish center's second goal of the season.
"We made a couple mistakes. They obviously had a push back and scored on their chances in the third," Zibanejad said.
"I thought we did a pretty good job up until the third but it wasn't a full 60 minutes and that's why we sit here with the loss."
Vancouver's Travis Hamonic played his first game of the season after starting the year on a temporary leave of absence to deal with personal matters. The 31-year-old defenseman can't travel with the team until he's fully vaccinated, a process head coach Travis Green has said will be completed "soon." ... Fox played his first game since agreeing to a seven-year, $66.5 million extension with the Rangers on Monday. ... Tuesday was the first time the teams have played each other since Jan. 4, 2020, when the Canucks won 2-1.
Rangers: At Edmonton on Friday night.
Canucks: Host Nashville on Friday night.
Canucks beat Rangers in wild OT finish
The Canucks rally from two goals down to beat the Rangers 3-2 in overtime.