Chris Kreider scored his fourth goal in six games in the first period when he converted on a feed from Mika Zibanejad while on a 3-on-1 that was similar to his power play goal Tuesday night against Vegas.
Michael Grabner appeared to double the led in the second period but the goal was waved off after Boo Nieves touched the puck with a high-stick.
The call proved to be a big one as Tampa tied the game with a shorthanded goal 11 minutes into the second.
Tampa had a goal of its own overturned early in the third period for goaltender interference, paving the way for overtime and Miller's goal.
Lundqvist made 27 saves in the win over what he called the best team in the NHL right now.
"We needed probably one of our better games this year to get two points," he said.
Lundqvist praised his teammates' ability to limit both Steven Stamkos and Nikita Kucherov, who were held off the scoresheet aside from eight shots on goal.
Vigneault was pleased with his team's play in all three zones. Like all coaches, he'll take the points and move on to the next one Saturday against the Panthers.
"I thought for our group it was good," he said. "It's a good win and we'll move on to Florida."