Bogut had season-bests and career-playoff highs of 14 points and 21 rebounds, and rookie Draymond Green added a career-high 16 points and 10 rebounds to power Golden State into the second round for the first time in six years.
The Warriors went ahead by 18 points early in the fourth before holding off one final Nuggets flurry.
Andre Iguodala scored 24 points and Ty Lawson had 17 in another disappointed early exit for Denver, which has lost in the first round nine of the past 10 seasons.
The Nuggets won 23 of their final 26 regular season games to earn the Western Conference's third seed, then lost four of six to the hot-shooting Warriors.
Golden State only outshot the Nuggets 40.3 percent to 34.7 percent in the finale but dominated the rebounding matchup again, 55-44. The Warriors overcame 21 turnovers — including 10 in the fourth — while the Nuggets only had seven.
Golden State will open the second round at San Antonio on Monday. The Spurs swept the Lakers in four games.
Warriors coach Mark Jackson ratcheted up the rhetoric after the Nuggets won a physical Game 5, saying Denver sent "hit men" out to take "cheap shots" at Curry.
The accusations earned Jackson a $25,000 fine from the NBA for an "attempt to influence the officiating," though neither side ever got out of control in the finale.
Except Curry's shooting — again
On the ball or off the dribble, Curry connected all over his home court in another dazzling second half. Curry swished all four of his 3-pointers in the third quarter and brought the announced sellout crowd of 19,596 roaring each time.
He also dribbled behind his back and threw his legs, crossing over Lawson and zipping a pass through traffic to Bogut for a thunderous dunk. With Denver starting to chase Curry around the perimeter, the spacing started to open up.
Curry added a one-handed pass off the dribble to a cutting Carl Landry, who finished for a three-point play over Wilson Chandler, flexed his arms above his shoulder and belted out a roar from the floor to give Golden State a 64-53 lead.
Green made a corner 3 in front of Denver's bench, and Klay Thompson hustled for a rebound and a quick put-back to put the Warriors up 80-62 with 9:11 to play.
The Nuggets failed to score for nearly six minutes during the stretch. But they rallied with 13 straight, held the Warriors scoreless for almost five minutes and gave the frenzied fans in Oakland reason to worry.
After the Warriors went back ahead by nine, they had 10 turnovers in the fourth and nearly collapsed. Chandler stole the rebound from Green for an easy layup to get within four, Iguodala hit another from long range and Kenneth Faried made 1 of 2 free throws to bring Denver within two with 32.4 seconds to play.
Iguodala stole Thompson's pass on the ensuing play. Chandler's running layup rimmed out, he missed a put-back and then knocked the ball out — a call that stood after a video review.
The Nuggets were forced to intentionally foul Jarrett Jack, who made both free throws to give Golden State a 92-88 lead with 7.3 seconds remaining to seal the victory.
The final buzzer sounded and gold confetti flew, with Green throwing his arms in the air — and running right to Jackson for a warm embrace.
Golden State seemed to have an answer for everything Denver threw its way in this series.
Even All-Star forward David Lee made a surprising return for the Warriors after not dressing the previous four games with what the team had called a season-ending hip injury. He grabbed one rebound and missed one jumper in 87 seconds off the bench in the first quarter.
No matter who played or when, the end-to-end action kept fans standing on their feet at Oracle Arena — nicknamed "ROAR-ACLE" by Warriors players in these playoffs — most of the way.
Lawson zipped in and out of the way with relative ease, twisting and turning for difficult layups and fallaway jumpers to carry the Nuggets to a 34-23 lead early in the second quarter.
With the crowd at full throat and mixing in chants of "de-fense!" and "Warr-i-ors!," players often couldn't hear whistles or even the buzzer.
Bogut's big first half — 10 points, eight rebounds, four blocks and two assists — kept the Warriors close, and they sliced Denver's lead to 42-40 at the break behind a late surge of shooting from their guards on the perimeter.
Faried, the target of Jackson's accusations of "dirty play" in Game 5, picked up his fourth foul with 10:23 to play in the third quarter. Three of those fouls came in the first 1:37 of the third before Nuggets coach George Karl took him out.
Soon after, Curry took control to send the Nuggets home for good.
NOTES: Aaron Hern, an 11-year-old from nearby Martinez, who was injured in the bombings during the Boston Marathon, attended the game with his family and honored on the videoboard during the first quarter. ... The Warriors entered 9-1 in home playoff games in which they had a chance to win the series since moving to the Bay Area prior to the 1962-63 season. ... Teams are 3-7 in closeout games in this year's playoffs.
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