And after each touchdown, fans tossed into the air some of the snow that was left over from an overnight accumulation of about 6 inches.
The Patriots began the day seeded third, but gained one of the AFC's two byes when they won after the AFC South champion Houston Texans lost and fell from first to third in the conference standings.
New England had trailed San Francisco 31-3 and Jacksonville 10-0 and sputtered at the start again on Sunday, turning the ball over on downs on its first possession and punting on its second.
Then it got a break and rolled the rest of the half.
Under heavy pressure, rookie Ryan Tannehill's pass was intercepted by Steve Gregory, who returned it 13 yards to the Miami 28. Gronkowski gained 19 yards on the first pass thrown to him. On the second play of the series, Brady connected with Welker over the middle for a milestone completion.
Brady moved into second place in NFL history with a touchdown pass in 48 consecutive games, breaking a tie with Hall of Famer Johnny Unitas. And Brady became the first player to throw at least one scoring pass in all 16 games for three straight seasons.
New England's top-rated offense got plenty of help from an improving defense.
Rookie Justin Francis had three sacks. Another rookie, Dont'a Hightower, recovered a fumble at the Patriots 1 on a botched handoff from Tannehill to Reggie Bush. On the ensuing drive, a pass to Welker popped loose and Miami's Chris Clemons came up with the ball and ran about 60 yards for what officials ruled a touchdown. But the call was reversed and the play was ruled an incomplete pass upon video review.
Just another play that went New England's way.
There were plenty in the first half when the Patriots had advantages of 279-106 in yards, 18-7 in first downs and 18:35 to 11:25 in time of possession.
Brady finished with 22 completions in 36 attempts for 284 yards. Tannehill was 20 for 35 for 235.
The shutout was the first for the Patriots since they routed the Tennessee Titans 59-0 on Oct. 18, 2009. The Dolphins were blanked for the first time since losing 16-0 to the Chicago Bears on Nov. 18, 2010.