The Patriots, who are idle next week before finishing the season at Carolina, have roared back from a 1-3 start and can clinch a playoff berth Sunday if either the New York Jets, Baltimore or Seattle lose.
Brady caught a 23-yard pass from running back Kevin Faulk on a trick play, setting up a two-yard scoring run by Smith that began a 20-point second quarter and boosted the Patriots to 10-5.
The offensive explosion delighted a sellout crowd of 60,292 which gathered for the final regular-season game at Foxboro Stadium, the oft-maligned facility that has been home to the Patriots for more than three decades.
"Each week, it seems like we're talking about different people," Patriots coach Bill Belichick said. "Somebody steps up and makes the plays that helps us win.
"And it was a great emotional day with the fans. It was a tremendous atmosphere out there and a thrill for me to be a part of it and a thrill for our football team. I thought they really played their hearts out today."
At halftime, 31 former and current Patriots took part in a ceremony to mark each year of the stadium's existence.
After winning their fifth consecutive game, many New England players exchanged high-five handshakes with fans while doing a victory lap around the field.
"It was pretty exciting," Brady said. "It was pretty exciting for the players. It was the first time I've ever seen something like that. Hopefully, we can keep it going. Hopefully, there will be more in the future."
"I think on behalf of the whole organization and certainly the players and coaching staff, we just wanted to show our appreciation for the last regular-season game scheduled here today," Belichick said.
The team will move next season into CMGI Field, located adjacent to the current home, but the Patriots greatly enhanced their playoff chances Saturday and have a chance of hosting an AFC postseason game in January.
The Dolphins (9-5) could have clinched a playoff berth with a win, but suffered their second straight loss. Miami was shut out last week in San Francisco, 21-0, and finish the season with home games against Atlanta and Buffalo.
"It seems like every time we got some opportunities to do something, it just didn't fall our way," Dolphins safety Brock Marion said. "In a game like this, you got to go out and take it, not to expect things to fall your way. We didn't do that today."
Brady completed 11 of 19 passes for just 108 yards, but one was a 23-yard touchdown strike to Patrick Pass that gave the Patriots a 14-0 lead.
The Dolphins had gone seven quarters without a touchdown before Jay Fiedler threw a 10-yard scoring pass to Jeff Ogden with 1:28 remaining. The Patriots recovered the onside kick and ran out the clock.
"I think we dug ourselves a hole in the first half and had some opportunities in the second," Miami coach Dave Wannstedt said. "I thought our defense came out in the second half and played good. It was obviously too late."
Miami's Olindo Mare kicked 36-yard field goals in the second and fourth quarters.
New England drove deep into Dolphins' territory on its first possession before turning the ball over on downs. It would be just a temporary setback for the Patriots, who took over the ball later in the quarter when Jed Williams fumbled near midfield after a reception.
Three plays later, Faulk took a direct snap fron center, rolled right and threw back across the field to a wide-open Brady, who was pushed out of bounds at the Miami two. Smith went over left guard on the next play.
On New England's next possession, Smith exploded for a 44-yard run to the Miami 29, setting up Brady's touchdown toss to Pass.
A fumble by Miami's Ed Perry on the ensuing kickoff led to Vinateiri's first field goal.
"We had a couple of breaks," Patriots cornerback Ty Law said. "We haven't yet played our best game, which is a good sign, so hopefully we can minimize the mistakes and play our best football from here on out."
Miami had several chances to get back in the game in the second half but failed to capitalize.
Lamar Smith had just 33 yards on 12 carries. He also fumbled near the goal line with 8:51 left after a jarring hit by safety Tebucky Jones.
The Dolphins played without leading receiver Oronde Gadsden, who has a hamstring injury, and Fiedler clearly missed him.
Chris Chambers caught seven passes for 124 yards and Fiedler was 21 of 37 for 320 yards. But many of those yards came after the the Dolphins dug themselves a huge hole.
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UPI Sports Results for September 20, 2014