FOXBOROUGH — Earlier in the season, when the Patriots were getting turnovers at a much slower clip than the 12 they’ve taken away in the last five games, some players, and even coach Bill Belichick, said the team just had to continue playing sound defense, turnovers or not. If the ball was out, that was a good sign, even if the Patriots weren’t recovering possession.

Not Malcolm Butler. Close wasn’t good enough.

Butler had something else in mind. Something more like the performance he pulled out of his stocking Saturday, when he intercepted two passes and recovered a fumble in the Patriots 41-3 win over the Jets.

“I’ve been hoping for a game like this. It finally came,’’ Butler said.

Butler intercepted Bryce Petty in the first quarter, batting the ball intended for Robby Anderson and catching it on the way to the ground. Then he made a heads-up play to recover the fumble forced by linebacker Elandon Roberts less than eight minutes of game-clock later.

Butler finished the hat trick at the end of the third quarter, picking off Ryan Fitzpatrick, who had replaced the injured Petty (shoulder), and running it back 21 yards.

Butler would take a performance like that any week of the season, but he sure doesn’t mind the timing. He redeemed one of his worst games of the season, Nov. 27 against the Jets, when he gave up a 40-yard pass and a touchdown to receiver Quincy Enunwa. Butler was in close coverage for both catches but didn’t get his hands on either pass.

“I just had the mentality to make sure I didn’t drop any balls this game and it didn’t happen,’’ Butler said. “I struggle with the Jets, man, there’s something about those Jets, but just glad to get the win.’’

He also shouldn’t mind that the stellar showing comes the week after he was left off the list of nominees for the Pro Bowl. Butler now has two more passes defensed (16) and two more interceptions (4) this season than he did all of last season, when he was selected to the Pro Bowl.

“If I would have made it, it would have been motivation, if I didn’t, it was going to be motivation, so either way I go it just works itself out,’’ Butler said.

The way Butler and his teammates have been playing, it might not have worked out for him to go to the Pro Bowl, anyway. The Patriots defense hasn’t given up a touchdown in more than two games, and had the Jets feeling so snakebitten they elected to kick a field goal while trailing by 41 points.

“You’ve got to take what you can get sometimes,’’ Butler said.

Now that his players have been coming up with the ball more often, Belichick acknowledged how pivotal those plays can be.

“There’s nothing that correlates more to winning than turnovers. Number 1 is points, No. 2 is turnovers,’’ Belichick said.

The Patriots have gotten at least one turnover in each of the last five games and, according to Butler, that means there should be more to come.

“Once you start doing things week in and week out, good things week in and week out, it starts to become contagious,’’ Butler said. “You know, the turnovers, not giving up points, just playing good football overall.’’

A little intra-squad competition helps. Cornerback Eric Rowe, who had the Patriots’ other interception Saturday, joked he wanted a second one to pull even with Butler.

“In my head I was like, ‘Man, he has two. I want more.’ We don’t really look at it like that, but in my head today I was like ‘Man I need one more so I can stick with him’,’’ Rowe said.

The two may be friendly rivals, but they’re also loyal teammates. Rowe said Butler should have made the Pro Bowl.

“He did get snubbed and he knows that, but he knows ultimately it’s kind of about what he’s doing for the team and obviously, he’s playing at a really high level right now, especially this last little stretch,’’ Rowe said.

“I mean, we talked about it and he was like ‘Man, I should have made it,’ but he’s like, ‘Whatever.’ Obviously he knows the team is more important. It’s kind of one of those — I think it’s like a popularity vote at the time. He says it’s whatever.’’

Nora Princiotti can be reached at nora.princiotti@globe.com. Follow her on Twitter at @NoraPrinciotti.