"This is the worst I've felt after a game," Golden State coach Mark Jackson said. "We got outworked tonight. They dominated us on the boards (and) we made critical mistakes down the stretch that are simple basketball plays to make. It's just a bad feeling."
Golden State had a chance to win in the first overtime leading 94-92, but Thompson missed a pair of free throws with 13 seconds left.
Both teams also missed several chances to win in regulation.
The Nuggets led 86-82 late in the fourth quarter, then went scoreless over the final two minutes, while the Warriors missed three shots in the final minute, including a long jumper by Stephen Curry as the buzzer sounded.
The wild finish was in stark contrast to the first half, when Denver appeared on the verge of a blowout.
The Warriors, already without 7-foot center Andrew Bogut, were even more limited in their front court after sending Jeremy Tyler home because of an illness. That left only rookie Festus Ezeli and Andris Biedrins to handle the middle against a Nuggets team that went into the night leading the NBA in rebounding, points in the paint and second-chance points.
The results were predictable — at least early.
Denver dominated inside early, outscoring the Warriors 16-0 in the paint in the first quarter to go along with five blocked shots. Iguodala had 11 points in the period, including an emphatic dunk with 26 seconds left to cap a 10-0 run and give the Nuggets a seemingly comfortable 24-13 lead.
Golden State, coming off a 24-point loss to the Los Angeles Lakers on Friday, chipped away in the second quarter behind Lee and Thompson, then pulled within 40-39 at the half after Richard Jefferson scored on an alley-oop dunk.
The Nuggets maintained their slim lead going into the fourth but couldn't put the Warriors away in regulation.
Miller added 15 points and eight assists for Denver, while Corey Brewer had 11 points.
Notes: This was the first of three games between Denver and Golden State over a 20-day period. After that, the two Western Conference rivals don't play one another again in the regular season. ... The Nuggets play 12 of their next 16 games on the road.