Miami's defense also came up with a goal-line stand in the third quarter and stopped the Colts four times inside the 10-yard line, including the final drive.
It was Miami's fifth straight win at the RCA Dome.
"The Colts are going to be one of the best offenses again in the league and to hold them to 13 points at home is a great accomplishment for our defense," Dolphins coach Dave Wannstedt said. "Don't underestimate what that means to us."
Upon hearing of the death of former Colts great Johnny Unitas, Manning expressed a desire to wear the black high-top cleats popularized by the Hall of Famer. But the NFL rejected the idea and threatened a $25,000 fine if Manning did not comply.
Manning dug the Colts a 21-3 hole with his three interceptions, including one deep in Miami territory late in the second quarter.
"It's tough when you spot a good team as much as we spotted them and made the kind of errors that allowed them to get ahead," Colts coach Tony Dungy said.
The Colts (1-1) had reduced their deficit to 21-13 when Manning took over on his 16-yard line with 2:15 to go. He moved the team downfield with the help of Edgerrin James, who ran three times for 30 yards on the drive. But the Dolphins kept James and his teammates inbounds, burning the clock.
Manning's 26-yard sideline pass to Troy Walters moved Indianapolis to the 12 with 19 seconds left, but Qadry Ismail was tackled inbounds after a six-yard catch. Manning had to spike the ball with two seconds left and his end-zone pass over the middle to Ismail was easily batted away by safety Brock Marion on the final play.
Williams carried 24 times as he became the first Dolphin to put together 100-yard rushing games since Bernie Parmalee in 1994. He also caught two passes for 62 yards and a touchdown. Fiedler completed 13 of 18 passes for 187 yards.
"I've been giving Norv (offensive coordinator Norv Turner) a hard time about not throwing me the ball," Williams said. "I had 60 catches last year (with New Orleans)."
Manning was 26 of 45 for 289 yards and a touchdown to Marvin Harrison, who caught 11 passes for 144 yards. James carried 30 times for 138 yards but was stopped four straight times from the two in the third quarter.
An interception and 20-yard return by David Macklin, an unnecessary roughness penalty on Miami and a 33-yard pass from Manning to Harrison moved the Colts to the two. James gained one yard on three carries into the line before trying to go left and getting dragged down for a loss by Tommy Hendricks.
"It was very frustrating," Manning said. "You kind of get stuck in that run mode because we gained a little each play until fourth down."
"I just tried to hit it up there, close the gap," Dolphins middle linebacker Zach Thomas said. "We've been playing good goal line and I'm glad we kept it going."
Manning engineered three long drives in the fourth quarter. The first went 80 yards and capped by his 16-yard pass to Harrison. The second was for 71 yards and stalled at the seven before Mike Vanderjagt kicked a 23-yard field goal with 2:54 to go. And the third carried 78 yards and ended with Marion's knockdown.
"That was the difference in the game," Dungy said. "We went up and down the field as much as they did. They scored touchdowns with theirs. We didn't score at all at times, had to settle for field goals."
The Dolphins scored on their first two possessions, with Derrick Rodgers' interception in between. Fiedler bootlegged in from a yard out before throwing a nine-yard touchdown pass to Rob Konrad for a 14-0 lead late in the first quarter.
In the second quarter, Vanderjagt kicked a 25-yard field goal after a drive died at the seven. Fiedler immediately responded with a 52-yard screen pass to Williams before finding the big tailback with a 10-yard touchdown throw and a 21-3 lead with 1:53 left.
"Even when they put eight in the box, we still ran the ball," Williams said. "In that situation, they really can't win. They bit on a lot of fakes because they had to respect the running game."
Patrick Surtain picked off Manning in the end zone in the final minute of the first half.