"I didn't know what his first name was," defensive tackle Jared Odrick said. "That's being thrown in the fire and reacting. That's a pure athlete."
The Dolphins (8-6) won their third game in a row to help their AFC wild-card chances. They also snapped a streak of seven consecutive losses to the Patriots (10-4), who missed a chance to clinch their fifth consecutive AFC East title.
Miami's breakthrough win left even stoic second-year coach Joe Philbin in a mood to celebrate.
"I might jump in my pool," he said. "I've been in it once since I've been here."
Brady threw for 364 yards but was frustrated by a succession of squandered scoring chances. He answered only two questions at his postgame news conference before cutting the session short.
"We didn't do a good job in the red area, and didn't do a good job finishing drives," he said. "We came up on the short end of the stick. It wasn't a good day. Just couldn't make enough plays."
The Patriots had mounted second-half rallies to win their past three games, and were on the verge of doing it again when Brady led them from their own 20 with 1:15 remaining to the Miami 14.
"The longest 75 seconds of my life," said Miami quarterback Ryan Tannehill, who squirmed on the sideline.
With New England out of timeouts, Brady threw for the end zone four times in a row. The first three passes went incomplete, including one intended for Danny Amendola that Thomas broke up on first down.
Then, Thomas stepped in front of Austin Collie to make his interception.
"I knew being the new guy that Tom Brady would come after me," Thomas said.