Reserve guard Blake Ahearn broke a tie when he made a 3-pointer from the left corner with 33 seconds to play to give the Timberwolves a 107-106 lead.
"We have a lot of options on that play, and Blake was one of those options," Minnesota coach Randy Wittman said.
Carter hit a free throw with 19.8 seconds left to tie the score.
Ollie, scoreless until the late free throws, went to the line for his first two attempts with 16 seconds remaining. He added his final two with 2 seconds to go.
Miller helped get the Timberwolves off to a strong start in the first quarter. He scored 11 points as Minnesota built a 31-18 lead with 2:47 left in the first quarter.
"That was probably the best start we've gotten off to all season," Wittman said. "We executed our offense well, and we were shooting the ball well."
Denver's Dahntay Jones started a 13-3 run with a field goal from the left wing at 1:39. The Nuggets tied the score at 39 with eight minutes to play in the second quarter on back-to-back 3s from Smush Parker.
Kleiza hit two free throws to give Denver its first lead, 42-41 with 6:48 left in the second quarter.
Sebastian Telfair's buzzer-beating 3 stretched the Timberwolves' lead to 57-51 at halftime.
"We've been playing very good defense most of training camp," Karl said. "This is probably our worst defensive game. Some of it is because of our shot selection and some of it was Minnesota played a very good game."
The Timberwolves shot 53 percent from the field and 78 percent (25-for-32) from the free-throw line. Denver shot 48 percent on field goals and 76 percent (26-for-34) on free throws.