Miller drove left past rookie Draymond Green, did an up-and-under between two defenders under the basket and banked the ball off the glass with his right hand.
The Warriors inbounded the ball and Stephen Curry's desperation 3-pointer wasn't anywhere close as the horn sounded and the Nuggets celebrated their 24th straight win at the Pepsi Center.
Miller scored 18 in the frenetic fourth quarter.
Game 2 is Tuesday night at the Pepsi Center, where the Nuggets posted an NBA-best 38-3 home record during the season.
"It could have went either way tonight so we have to be better prepared on Tuesday," Miller said.
The Warriors trailed 93-92 when Curry was pickpocketed by Ty Lawson, whose layup made it a three-point game with 35 seconds remaining.
Curry, who shot his way into NBA history by sinking a record 272 3-pointers this season, got a whistling pass from Jarrett Jack and swished a contested 3 with Lawson all over him, then did his funky dance downcourt during the timeout with 14.5 seconds left.
Miller inbounded the ball, then got it back and isolated on Green until the driving to the hoop for the winning bucket.
"We found a way to pull it out," Miller said. "This was a tough game."
The sixth-seeded Warriors, playing in their first playoff game in six seasons, lost All-Star David Lee to a possible right hip flexor strain in the fourth quarter when he banged into JaVale McGee. He shot both free throws, then couldn't run back downcourt and was taken for X-rays.
Lee finished with 10 points and 14 rebounds, all but two off the defensive glass. Klay Thompson led Golden State with 22 points and Curry had 19. Jarret Jack added 10 points and 10 assists.
Center Andrew Bogut, who's been bothered by a bone bruise on his surgically repaired left ankle, had 14 boards and nine points for the Warriors.
Corey Brewer sparked a 13-2 run that the Nuggets used to take a 71-64 lead into the fourth quarter, but the Warriors responded to Lee's injury by quickly erasing an eight-point deficit.
Then Miller took over. Trying to make it out of the first round for the first time in his 13-year NBA career, he came up with several big baskets in the final period.
The frenetic finish followed a plodding first 2½ quarters that wasn't anything like the high-octane, run-and-run game everyone expected from these two teams. That notion fizzled in the first half with a rash of poor shooting, too many turnovers and a bunch of debatable calls. Denver would have matched its lowest point total for a first half if not for Wilson Chandler's breakaway basket at the buzzer that pulled the third-seeded Nuggets to 48-44 at halftime.
In their three home losses this season, the Nuggets managed 42 first-half points against Miami on Nov. 15, 43 against Minnesota on Jan. 3 and 50 against Washington on Jan. 18, before winning their last 23 home games heading into the playoffs.
"You play 82 games to get to this spot, to have your home court help you with confidence, help you with the jitters," Nuggets coach George Karl said before tip-off. "I think Golden State is a very good opponent and a very difficult opponent in a lot of ways. I don't expect it to be anything but a damn good basketball series. Today's the first game and let's see what happens."
What happened was the Nuggets nearly lost the home-court advantage they so cherished.
Not having their best free-throw shooter in Danilo Gallinari (out with an ACL injury) and their top rebounder in Kenneth Faried (ankle) didn't help.
But the Nuggets pulled out another close one at home.
They haven't lost at home in more than three months, but Golden State coach Mark Jackson dismissed any notion that it was because of altitude or boisterous crowds.
"We've watched every bit of film and I haven't seen the altitude score and I haven't seen a fan score or rebound," Jackson said before the game. "At the end of the day, credit goes to that coaching staff led by George Karl and that group of players. Win, lose or draw, it won't be because of the fans or the altitude."
Jackson played in plenty of playoff games but this was his postseason coaching debut. He said he had no butterflies, however: "We're playing with house money. We're going to roll this dice and have fun with it. Nobody expects us (to win) and that's OK. If I was picking, I'd pick them to win it. I'd pick them to win this series. I'd give them the edge at coaching, too."
Karl laughed at that remark.
"Pile on those expectations," he said. "Let's go, baby!"
Karl freely admitted he was a bundle of nerves before the game: "I'm nervous for summer league games, so why wouldn't I be nervous for this game?"
NOTES: In the first quarter, McGee appeared to have a clean block when he snatched Carl Landry's short jumper out of the air at its apex but was whistled for goal-tending to his dismay. ... The Nuggets won three of four games in the regular season.
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