Williams ran for 111 yards and two touchdowns on 20 carries and Edwards also scored twice as the Dolphins posted their 11th straight season-opening victory, a 49-21 rout of the Detroit Lions.
It was also Miami's 16th straight win in September since 1993.
The 49 points were the fourth most scored by the Dolphins in team history and the most they tallied in a contest since scoring 52 against the New York Jets on Sept. 3, 1995.
"I thought our guys really came out prepared today," Miami coach Dave Wannstedt said. "We had no turnovers on offense and ran the ball and played the run pretty good. I was real happy with the effort and execution of our football team today. It was a good start."
Williams opened the scoring with 3:07 left in the first quarter when he took a handoff from Jay Fiedler and went over the top for a one-yard touchdown, one play after fullback Rob Konrad fumbled a completed pass out of bounds just outside the goal line.
Williams, who is the first Heisman Trophy winner to ever play for the Dolphins and was acquired from New Orleans in the offseason, added to the onslaught in the third when he ran left for a two-yard score, making it 35-7.
"The line did a great job," Williams said. "There was room to run and I tried to get the most out of each play that I could."
"Guys like (Williams) are able to see things and find holes where maybe a lot of people think there aren't," Miami center Tim Ruddy said. "I think he's certainly in that category."
Edwards, who was signed as a free agent in the offseason after missing the last three campaigns because of complications from a dislocated knee, scored his first touchdown with just over 11 minutes left in the game when he caught a four-yard pass from Fiedler to give the Dolphins a 42-14 lead.
With 4:21 left to play, Edwards scored on a one-yard run to cap the Dolphins' scoring.
"It felt great. I was just thankful I was in this position," said Edwards, who signed with Miami after being waived by New England. "The preseason touchdown was special but this one counted. I feel like a rookie again. I'm excited to be here. It's definitely another goal I've accomplished."
The 28-year-old back ran for just 20 yards on four carries but caught four passes for 38 yards.
"I was delivered a miracle to be where I'm at," said Edwards, who rushed for more than 1,000 yards as a rookie in 1998 before suffering a horrific knee injury in a flag football game at the Pro Bowl. "This is the best. Everything's hitting right on point with me, Ricky and Travis (Minor). We're going to be hard to stop."
Minor ran for 45 yards on 11 carries.
"I'm surprised at a lot of things today," Detroit coach Marty Mornhinweg said. "First the score, second that they were able to run the football as well as they did. "I thought we would have success especially against the running game."
"I wouldn't really say Ricky was a big difference," said Lions safety Brian Walker, a former Dolphin. "It all starts up front and if those guys are blocking like they were today, a lot of guys will have big days."
The Dolphins are 6-2 all time against the Lions and have won all four contests at home.
The Lions got on the board with just over four minutes left in the first half when quarterback Mike McMahon scored on a two-yard run to complete a 14-play 80-yard drive.
But the Dolphins responded on the next drive when they marched 66 yards in just 88 seconds as Fiedler hit Chris Chambers for a nine-yard touchdown, making it 28-7. Edwards caught two passes on the drive for 28 yards to set up the touchdown.
"We were having a lot of fun out there," Fiedler said. "We were executing and it seemed like everything was working well."
Fiedler threw for 207 yards, completing 18 of 27 passes, but gave most of the credit to Williams.
"I can use everyone on the field," Fiedler said. "I don't have to zero in on one person. That puts a lot of pressure on the defense.
The Lions added two scores in the second half. McMahon completed a 13-yard pass to running back Lamont Warren in the end zone with 8:41 left in the third, making it 35-14, and fullback Cory Schlesinger capped the scoring with a two-yard run with just under seven minutes left.
"We gave up 21 points on defense but quite frankly, I don't think we should have given up any," Dolphins defensive end Jason Taylor said. "Ricky Williams pounded the ball. Give credit to him and the line for making those yards."
McMahon was 13 of 29 for 179 yards, including an interception. He was replaced late in the fourth quarter by rookie Joey Harrington, the Lions' first-round pick. Harrington was four of 11 for 41 yards.