The much-maligned Feeley threw for the game-winning score, a 21-yard touchdown pass to Derrius Thompson on fourth and 10 with 1:23 left.
The stunning victory rivaled the biggest in Dolphins history, because they've rarely been so woeful. With the win, Miami (3-11) avoided being swept by AFC East opponents this season.
"I keep asking everybody, 'Is it real?' " said linebacker Brendon Ayanbadejo, who made the pivotal interception. "It's a big-time win. I can't believe it."
New England (12-2), which has already clinched the AFC East, lost ground to Pittsburgh (13-1) in the race for home-field advantage. The defeat was just the second in 29 games for the defending Super Bowl champions.
The Patriots appeared to clinch the win when Brady threw his third touchdown pass, a two-yarder to Daniel Graham for a 28-17 lead with 3:59 left.
"That's a situation we're usually pretty good in," Patriots Coach Bill Belichick said. "We weren't tonight. It wasn't about intensity. It was about execution."
The Dolphins drove 68 yards to score on Sammy Morris's one-yard run with 2:07 remaining, and then came Brady's uncharacteristic mistake.
On third down, he threw as he was being sacked by Jason Taylor. The ill-advised, off-balance pass sailed directly to Ayanbadejo, giving Miami a first down at the Patriots 21 with 1:45 to go.
Three plays later, on fourth and 10, Thompson beat Troy Brown -- a receiver who doubles as a fifth cornerback -- and made the winning catch in the end zone.
"We've been on the losing end of so many drives like that," Feeley said. "It feels good to come out on top."
Brady's fourth interception, a desperation throw picked off by Arturo Freeman with 58 seconds left, sealed the upset. The takeaways offset a 121-yard rushing performance by New England's Corey Dillon.
"We really didn't play that well defensively," Taylor said. "But we made enough plays to win the ballgame. That's all that matters."