This time, the New England defensive tackle didn't commit a penalty to give the Jets kicker another chance.
Jones blocked Folk's 58-yard attempt on the final play and the Patriots had just enough offence with Tom Brady's three touchdown passes to edge surprisingly competitive New York 27-25.
"It's good to feel that ball hitting my hand and it's good to get the win," Jones said. "I just played it legally and did all that I could."
Last season, Folk was wide left on a 56-yard attempt in overtime, but Jones was penalized under a new rule against players pushing a teammate into the opponent's formation. Folk then connected from 42 yards, giving the Jets a 30-27 victory.
"After what happened last year, I thought it was fitting that he made that play," Patriots coach Bill Belichick said.
Folk's longest field goal in his career travelled 56 yards, but he had made all 13 of his attempts this year so the 58-yarder was makeable.
"It felt pretty good off my foot," he said. "I did kick it low, in order to give it enough to get it there."
But when Jones batted it down, the Patriots (5-2) raced onto the field in celebration of a win that was much tougher than expected.
The Jets (1-6) suffered their sixth straight loss, the longest streak in Rex Ryan's six seasons as coach, after taking a 19-17 lead with just under 9 minutes left in the third quarter on Chris Ivory's 1-yard run.
Stephen Gostkowski put the Patriots ahead to stay with his second field goal, a 36-yarder with 4:10 to go in the third. Brady's 19-yard touchdown pass to Danny Amendola gave them an eight-point lead with 4:10 remaining.
The Jets made it 27-25 on Geno Smith's 10-yard scoring pass to Jeff Cumberland with 2:31 left, but Smith's pass for the 2-point conversion was incomplete.
Ivory finished with 107 yards rushing.
"This is extremely frustrating," Smith said. "I felt it was going to happen for us."
Smith, last in the NFL in passer rating, had a solid game with 20 completions in 34 attempts for 226 yards and no interceptions.
Brady was 20 for 37 for 261 yards and no interceptions.
Each team played without two key starters who went on season-ending injured reserve after being hurt Sunday — cornerback Dee Milliner and left guard Brian Winters for the Jets and linebacker Jerod Mayo and running back Stevan Ridley for the Patriots.
"As a whole offence we needed to step up," running back Shane Vereen said. "You can't replace Stevan."
The Patriots led 17-12 at halftime despite having the ball for just 7:57, while the Jets held it for 22:03.
"When the defence is out there as much as they were tonight it's going to be tough for them," Brady said. "We've got to help them out."
The Jets got inside the New England 30 on all four of their first-half possessions. But they managed just four field goals by Folk, covering 22, 47, 46 and 27 yards.
The Patriots went three-and-out on two of their possessions in the half but made the most of the other two as Brady threw touchdown passes to Vereen — a 49-yarder on their fourth offensive play and a 3-yarder that made it 14-9 with 4:22 left in the half.
After Folk's fourth field goal, the Patriots got the ball with 55 seconds remaining in the half. On third-and-10 at the New York 44, Antonio Allen was called for defensive pass interference on Amendola, putting the ball at the 12.
An unnecessary roughness penalty against guard Jordan Devey pushed the ball back to the 27. After a 6-yard run by Vereen, Gostkowski kicked a 39-yard field goal on the last play of the half.
The Jets were competitive against a top team for the second straight game. They lost to the Denver Broncos 31-17 on Sunday.
"The result is not showing on the scoreboard, but I think we are getting there," Ivory said.
NOTES: The Jets outgained the Patriots on the ground 218 yards to 63. ... Former Patriots cornerback Ty Law was honoured at halftime for his induction into the team's Hall of Fame. ... Amendola had four catches in his first six games but came up with a big one then watched his team hold the lead. "I almost had a heart attack at the end," he said, "but the defence came up huge." ... Brady got his 41st straight regular-season home victory against an AFC opponent.
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