Saints claim first place in the NFC South with a 31-15 win at the Chicago Bears _lowres

New Orleans Saints quarterback Drew Brees (9) celebrates as he leaves the field after an NFL football game against the Chicago Bears Monday, Dec. 15, 2014, in Chicago. The Saints won 31-15. (AP Photo/Charles Rex Arbogast)

CHICAGO — It’s been a crazy, unpredictable, hard-to-define, upside-down cake of a season for the New Orleans Saints.

But there are worse things than coming into Week 15 of the regular season 5-8. There are the Chicago Bears.

In the latest episode of “Chicago No Hope,” Bears coach Marc Trestman is facing withering criticism for steering a rudderless ship of a franchise. Offensive coordinator Aaron Kromer is facing criticism for criticizing quarterback Jay Cutler.

Cutler is making it look more and more like the Bears will suffer whatever price they have to pay to dump his overpriced behind in the offseason. And defensive coordinator Mel Tucker (who coached defensive backs at LSU in 2000) looks like he may be on the way out of town first.

“There are some problems in Chicago right now,” said ESPN’s Jon Gruden, his currency being the understated phrase.

Into this morass rode the Saints on Monday night, lugging some hefty baggage of their own.

They came braced by the sobering reality that their margin for error had finally eroded completely away, thanks to their stunning four-game home freefall and the Carolina Panthers’ equally stunning two-game winning streak to, ahem, ascend to a 5-8-1 record.

This was the Saints’ shark week. Finally it had come down to eat or be eaten. The Saints still controlled their own destiny, yes. Win their last three games, and they would be NFC South champions. Atlanta (5-9) and Carolina could go kick themselves for all the opportunities they’ve missed this season.

But first the Saints had to gain control of a season, of a fate that has always raced just ahead of their grasp. So naturally with their home field in the Mercedes-Benz Superdome suddenly so inhospitable, they went on the road for a third straight victory.

A chilly, misty rain with temperatures in the mid-40s greeted New Orleans as it arrived at Soldier Field.

Saints weather? Not for the team whose home box score always reads “72 degrees, indoors.” But the Saints took to the cold and clammy conditions like a St. Bernard.

They started the No Pity for Cutler Party early. With safety Kenny Vaccaro on the bench in the Saints base defense, and with a switch at linebacker (Parys Haralson starting for Junior Galette and Ramon Humber starting in Haralson’s place), Patrick Robinson pulled in a tipped interception and returned it to the Bears 24. But the Saints came up empty as the star-crossed Nick Toon fumbled a Drew Brees pass at the Bears 3.

In earlier games this season, such a missed opportunity would have haunted the Saints all the way to another maddening defeat.

This time instead of being tormented, the Saints came back fighting. New Orleans opened the second quarter with an 8-yard Brees to Josh Hill touchdown pass. The Saints closed it with a 9-yard Brees to Marques Colston scoring strike, proving again that if the Saints can do one thing consistently well this season it’s execute the two-minute drill.

Another Brees-to-Hill TD pass on the first drive of the second half made it 21-0, then it was 24-0 after a Shayne Graham field goal. The Bears came back to make it 24-8 on a Cutler TD pass and two-point conversion, leaving not visions of sugar plums but the Debacle in Detroit dancing in the heads of Saints fans the world over.

But there was little reason to fret. The Saints defense, shaken and stirred to aggressive life at last, went on the offensive time and again, leaving Cutler with a pre-Christmas nightmare of a night:

Interception, sack, sack, sack, interception, sack, interception, sack, sack and sack.

The Saints offense went into some sort of hibernation as the Bears mounted their threat, but it turned out to be much ado about not much. Mark Ingram’s 15-yard touchdown run with 1:47 left was the final touch on what turned out to be a 31-15 victory.

Ingram’s sprint gave New Orleans just 83 rushing yards, a low total for the Saints in a season when running the ball consistently well has been one of the strengths that have flown in the face of so many defeats.

But against a Bears defense as porous as the Saints defense has been, on a night when the conditions were far less than ideal, Brees was brilliant. He was 18-of-20 at halftime and finished 29-of-36 for 375 yards and three touchdowns. Brees stepped up and around the Bears pass rush most of the night, and his throws were unfailingly on the mark.

Where does this win leave the Saints? In first place, for a start. Now they go home, a surprisingly dicey proposition, to host the equally hard-to-figure Falcons before closing the regular season at Tampa Bay.

“We’ve shown a couple of times we can respond to adversity,” Brees said. “Let’s see if we can handle success, too.”

It was just one win over a bad team, but the Saints have had mistakes by a Great Lake (I’m thinking of you, Cleveland) that didn’t turn out nearly this well.

If they Saints were in the NFC North like the Bears, their season would be over.

But they’re not, and it’s not. It hasn’t been beautiful, but the Bears would gladly make a trade — and maybe throw in Cutler to boot.

Follow Scott Rabalais on Twitter: @RabalaisAdv.