Favre was able to start the 292nd straight game of his career despite two broken bones in his left ankle, but took a big hit late in the fourth quarter from Myron Pryor and was carted off the field sporting ice on his jaw.
Tarvaris Jackson came in and threw a touchdown pass to pull Minnesota within three, but a lengthy drive, capped by a Green-Ellis TD run, gave the Patriots (6-1) their fifth straight win.
Tom Brady completed 16-of-27 passes for 240 yards with a touchdown to Brandon Tate, who logged 101 yards on three receptions in the victory.
Adrian Peterson ended with 92 yards and a score on 25 carries, while Favre was 22-of-32 for 259 and an interception before exiting for the Vikings (2-5), who are tied with the Lions for last place in the NFC North after their third loss in four games.
Randy Moss, in his return to Gillette Stadium following the startling trade from New England to Minnesota earlier this month, was held to one catch for eight yards.
New England, holding a 21-10 lead, punted in enemy territory four minutes into the final stanza, and a pass interference penalty on Brandon Meriweather a little later gave Minnesota 1st-and-goal at the Patriots nine-yard line.
On third down, Pryor busted through the line and hit Favre up high just as he threw incomplete. The Pats were whistled for illegal contact to extend the drive, but Favre was helped to the sideline after the crushing tackle.
Jackson, in his first action of the year, connected with Naufahu Tahi out of the backfield for a one-yard score with 7:26 to go. Percy Harvin hauled in a two-point conversion in the left corner of the end zone, making it 21-18.
The Patriots sealed it with a 13-play, 80-yard trek that took 5 1/2 minutes off the clock. Danny Woodhead's 16-yard catch-and-run on 3rd-and-12 from the New England 44 highlighted the drive, and Green-Ellis was able to cross the goal line from two yards on the first play after the two-minute warning.
The Vikings opened the scoring with a six-minute drive that spanned the first and second quarters. On the first play of the latter frame, Peterson finished off a 12-play, 76-yard drive with a one-yard plunge. Replay was used to determine if the running back crossed that goal line, and the TD was upheld.
The Pats answered on their next drive, as Woodhead went in from three yards out on a direct snap 4:35 into the second quarter. Madieu Williams nearly had an interception on the scoring march, but the ball went right through his hands and into Tate's for a 32-yard gain.
Peterson was stuffed on 4th-and-goal from the one at the end of the half, but Ryan Longwell gave the hosts the lead again via a 24-yard field goal 5 1/2 minutes into the third quarter.
Shortly thereafter, Brady bought time with his legs and hit Tate in stride down the left sideline just past midfield. Tate did the rest, going 65 yards for a 14-10 lead near the eight-minute mark of the third.
After trading punts, Devin McCourty picked off a Favre pass and returned it 37 yards to the Minnesota 37. From the 13, Green-Ellis broke an arm tackle and went in for a touchdown with 2:25 left in the third.
The Patriots have won 16 consecutive home games against the NFC...The Vikings sent the Pats an undisclosed draft pick for Moss after the wideout was held without a reception in New England's Week 4 win over Miami...Harvin caught six passes for 104 yards...The Patriots play at Cleveland next week, while the Vikings try to get back on track at home against the Cardinals.