The win gave Miller 235 career victories, breaking a tie with six-time Vezina Trophy winner Dominik Hasek for the most in franchise history.
"I played with great teams and many great players, so this is very satisfying," said Miller, a fifth-round draft pick in 1999 who debuted for the Sabres during the 2002-03 season. "And it was a good comeback win for us. We played a solid game, sticking with it and getting chances."
Buffalo's Paul Gaustad tied it 3-all midway through the third period to send the game to overtime. The Islanders had a power play in the extra session when the Sabres were called for a bench minor, but New York couldn't get past Miller.
The 31-year-old netminder made 35 saves, including several key stops in overtime and the shootout.
Pominville, the Sabres captain, was especially gratified for Miller.
"He gives us a chance every night. We're excited and happy for him because he's an elite goalie," Pominville said. "I'm glad he was rewarded tonight."
The Sabres beat the Islanders for the first time in three meetings this season, preventing New York (21-22-8) from reaching the .500 mark for the first time since Oct. 22.
It was the Islanders' first shootout at home this season. Nielsen and P.A. Parenteau failed to score for New York in the shootout.
"With a 3-1 lead, we have to bear down more," Parenteau said.
"It's unfortunate we couldn't come out with two points," Montoya said. "Those were huge points on the line."
The Islanders are still 5-1-2 in their last eight and 10-5-2 in their last 17 contests. New York beat Carolina 5-2 on Tuesday night and topped Ottawa 2-1 on Friday night in overtime. Both wins came on the road.
Buffalo (22-24-6) has been beset by injuries all season. The Sabres played without leading scorer Thomas Vanek, who missed the game with an upper-body injury.
Nielsen opened the scoring with his eighth goal of the season at the 1:39 mark of the first after the Sabres gave away the puck behind their own net.
Nathan Gerbe tied it at the 5:32 mark with his fifth goal before Bailey beat Miller -- on an assist from Nielsen -- with about 4 minutes to go in the period to put the Islanders ahead 2-1.
Bailey's sixth goal of the season came just as a high-sticking penalty to the Islanders' Nino Niederreiter expired.
Nielsen knocked in his own rebound with 2:43 to go to make it 3-1.
"We just can't sit back, we have to play better," Nielsen said.
Islanders coach Jack Capuano gave credit for the comeback to Buffalo's stingy play.
"They had good puck protection," Capuano said.
The Islanders, who are 16-3-5 when scoring first, beat the Sabres 2-1 at Buffalo Nov. 29 and 4-2 at home Jan. 14. The teams are tied for 12th place in the Eastern Conference with 50 points, 10 points behind Toronto for the eighth and final playoff spot.
Miller smiled as he mulled his milestone, which comes amid a difficult season for the Sabres.
"All-time Sabres winningest goalie, I am proud of that," he said. "It's exciting to come in as a young player and achieve it. It will always be a source of pride."
Islanders D Travis Hamonic was hit in the face with a deflected slapshot late in the third period. Capuano said Hamonic was taken to a hospital for observation. ... Islanders C John Tavares led the league with 22 points (nine goals, 13 assists in January) becoming the first Islander to be first in the league in points in a given month since Bryan Trottier had 28 points in November 1980 ... Sabres coach Lindy Ruff has 548 wins with Buffalo. Only former Islanders coach Al Arbour (740) has more victories with one NHL team ... Buffalo has only two players (Pominville and goalie Jhonas Enroth) who have dressed for every game this season.