Miami improved to 8-3 and took over the top spot in the division by itself when the New York Jets fell to New England. Denver slipped to 6-6.
"It's good that we are getting it done in the fourth quarter," Dolphins linebacker Zach Thomas said. "We still haven't put together a full game. We are not yet in sync, but we're still 8-3."
Playing in front of a sparse crowd at Pro Player Stadium, the Dolphins were less than 14 mintues away from their second straight shutout loss at home.
But Jay Fiedler threw an 11-yard touchdown pass to rookie Chris Chambers 1:15 into the fourth and Mixon went untouched with his interception to give Miami a 14-10 lead with 9:51 remaining.
Denver then fumbled the ensuing kickoff and Miami turned the miscue into a one-yard touchdown run by Lamar Smith with 8:59 remaining.
"I think that is a great example of why we won this football game," coach Dave Wannstedt said. "When we were struggling on offense early, the defense came up with a big play and the special teams came up with a big play."
After Smith's touchdown, the Dolphins' defense did the rest, shutting down Brian Griese in his return home. The Broncos' quarterback is the son of former Miami Hall of Fame quarterback Bob Griese.
"It brought back memories," said Griese, who threw for just 135 yards. "But that goes away in a hurry once the game starts."
Denver running back Terrell Davis returned from a two-game absence and looked strong, rushing 20 times for 97 yards. But the Broncos, playing without star wideout Rod Smith, took a hit in their quest for a postseason berth.
"They found a way to win the game," said Denver coach Mike Shanahan, whose team is in the thick of the AFC wild card race. "There's not much else to say. They made the plays in the fourth quarter."
Neither team found the red zone until the first possession of the second half, when Griese completed a 13-play, 75-yard drive with a four-yard pass to tight end Desmond Clark.
The Dolphins finally got going on their second possession of the half, moving 58 yards on seven plays and cutting their deficit to 10-7 with 13:45 to play on Fiedler's scoring pass to Chambers.
Denver then moved the ball into Miami territory, but Mixon trapped Griese's short pass over the middle against his left leg and went untouched for his first career touchdown and a 14-10 lead.
It also was the first interception return for a touchdown by a Dolphins defensive lineman since Kim Bokamper in 1983.
"It's one of those plays you visualize, but it never happens," the 6-4, 285-pound Mixon said. "It was a great feeling to get into the end zone."
"I really didn't see what happened after I threw it," Griese said. "If I take a sack, we probably win that game."
Chris Cole fumbled the ensuing kickoff return at his own seven-yard line and Smith found the end zone two plays later to cap the scoring.
"I was trying to make something happen, but there's no excuse for turning the ball over," Cole said. "I cost this football team the game."
The Dolphins lost the possession battle by more than nine minutes and had only 218 yards to 273 for the Broncos, but they forced two of the game's three turnovers.
"We won the turnover battle, and when we had opportunities, we scored touchdowns instead of kicking field goals," Wannstedt said. "It wasn't a pretty win, but it was a great win."
Fiedler protected the ball and played consistently, completing 18 of 28 passes for 176 yards. Lamar Smith had only 36 yards on 15 carries but added five receptions for 43 yards.
"I was so pleased with his performance," Wannstedt said of Fiedler. "This is two games in a row that he didn't force anything or throw an interception."
Griese never got going for Denver, completing 18 of 33. He has thrown for fewer than 200 yards in each of his last three games, including a pair of losses.
The quarterback had few options. With wide receiver Ed McCaffrey already out for the season, Rod Smith was sidelined with a sprained ankle. He leads the NFL with 82 receptions.
Davis, who has had surgery to both knees since training camp, ended up with his best rushing total since he gained 101 yards in a season-opening win over the New York Giants.
"The legs got heavy and I came out a lot today," he said. "I was running out of wind. But as the weeks go by, I'll get better."
Cole and Karon Coleman -- who entered the game with three career receptions combined -- had four grabs each for the Broncos.
Denver scored the only points of the first half on a 45-yard field goal by Jason Elam with 7:01 left.