EAST RUTHERFORD, N.J. — Going into Sunday’s game with the New York Giants, the New Orleans Saints had painted themselves into a corner with no margin for error.

None whatsoever.

So when they committed some mistakes that were uncharacteristic of them in the first 10 weeks of the season, the Giants took every advantage of the opportunities handed them on a dreary Sunday in MetLife Stadium.

A much-needed 52-27 victory helped the Giants (8-5) maintain a one-game lead in the NFC East over the Washington Redskins and Dallas Cowboys, who both won shortly before they took the field.

At the same time, the Giants all but extinguished the final hopes of the Saints (5-8) to make the playoffs. They’re three games behind the Chicago Bears and Seattle Seahawks — the NFC wild-card leaders — with only three games remaining.

Even though they’re not out of it mathematically, the Saints know that making the playoffs won’t happen for the first time since 2008.

“It’s been a while since we’ve been eliminated from the playoffs,” Saints weakside linebacker Jonathan Vilma said after their third consecutive loss. “This is tough. … This is tough to swallow.”

Which makes the way they lost Sunday even worse, considering they allowed a club-record 287 kickoff return yards — including David Wilson’s 97-yard touchdown return for the Giants’ first score — and gave it away four times.

In their first 12 games, the Saints were among the NFL’s best at covering kickoffs in allowing 22.0 yards per return.

And after turning the ball over just 12 times in their first 10 games, they have given up 11 in the last three with Drew Brees throwing two more interceptions Sunday to give him nine in the three-game stretch.

“That was an embarrassment today,” Saints interim coach Joe Vitt said. “It starts with me. That’s where the blame should go. I didn’t do a good enough job of getting this team prepared.

“Here’s the key,” he said. “We can’t have four turnovers in a game, especially on the road against a good football team. We can’t allow them to have the returns that they had and score touchdowns on returns and have the chance to win on the road against a good football team.”

The Giants turned the four takeaways into 17 points, helping them on a 38-14 run over the final 301/2 minutes after the Saints trimmed a 14-7 deficit to 14-13 with just under two minutes left in the first half.

Until the Giants marched down the field to a 5-yard TD pass from Eli Manning to Domenik Hixon, the Saints were hanging in there.

A 73-yard interception return for a touchdown by cornerback Elbert Mack, which was offset moments later when Wilson slashed through the kickoff coverage unit, and field goals of 32 and 25 yards by Garrett Hartley had them right where they wanted to be.

The Giants had just 90 yards in total offense, which included a 35-yard drive that was capped by Manning’s 6-yard TD toss to Martellus Bennett, before they mounted their final drive of the first half.

After that, the floodgates opened.

On the Saints’ first play of the second half, Brees threw a pass that went through the hands of tight end Jimmy Graham and was picked off by Giants safety Stevie Brown — the first of his two second-half interceptions — and retuned 21 yards to the Saints’ 20.

Hat led to another short field against the Saints’ defense, which Manning converted into a 6-yard scoring run by Wilson.

Manning got the third of four scoring passes on the next possession.

He found Hakeem Nicks for a 25-yarder to make it 35-13 before the Saints rallied back with a 13-yard TD run by Darren Sproles and a 9-yard scoring pass from Brees to Sproles.

But the Saints, who had already given up kickoff returns of 58, 97 and 52 yards to Wilson, were hit by Jerrel Jernigan’s 60-yard return to set up a back-breaking 10-yard TD from Manning to Victor Cruz.

Another interception by Brown and 70-yard return set up a 39-yard field goal by Lawrence Tynes and all but finished off the Saints before Wilson had a 52-yard touchdown run with just over five minutes to play.

Wilson, a rookie, rushed for 100 yards and two TDs on 13 carries and added 227 yards on four kickoff returns for 327 all-purpose yards.

“I think our coverage unit has done a great job all along,” Vitt said. “We’re one of the top coverage units in the NFL. Today, all of a sudden, they return the first kickoff (for 58 yards) and the second kickoff for a touchdown — and we’re all stunned. We’re in damage control mode and the horse is out of the barn.”

Brees, who threw five interceptions in the Saints’ previous game against the Atlanta Falcons on Nov. 29, had another rough day.

He completed just 26 of 43 passes for 354 yards with one touchdown and the two interceptions. He was also sacked once and had a passer rating of 75.1.

“When you’re losing the way we’re losing, you’re a fragile football team,” a solemn Vitt said. “We have a lot of new guys, a lot of new faces on this football team.

“We have enough guys that are used to winning and used to doing things the right way. Unfortunately, right now, we don’t have enough of them.”