Saints quarterback Drew Brees, who came into this game having completed over 80 percent of his passes, finished completing 56.2 percent of his passes on an 18-of-32 showing for 217 yards.
Brees did not throw a touchdown pass for the first time this season and for the first time since a 47-10 win over Buffalo last season.
The Giants started out good enough, taking their opening drive 75-yards on 10 plays before capping it with a 2-yard touchdown pass from Eli Manning to receiver Sterling Shepard, who caught all 10 of his pass targets for a team-leading 77 yards.
The Saints began to chip away at the Giants' lead with four field goals by kicker Will Lutz, which gave them a 12-7 lead at the half before running back Alvin Kamara broke open the game in the third quarter with three touchdown runs.
The Giants' offense, meanwhile, went downhill. They did not score again until Aldrick Rosas nailed a 33-yard field goal with 2:02 in the third quarter. In between their opening scoring drive and the drive just before Kamara's first touchdown, the Giants offense managed only 62 yards of offense.
"Tonight was a frustrating one," Manning said. "We got off to a good start and had a good drive. We were moving the ball and making plays on this team, but we just weren't able to do it."
Perhaps the biggest problem for the Giants' offense was that they failed to attack the Saints' defensive secondary, which was banged up with injuries and deployed single-high coverage against receiver Odell Beckham Jr.
Manning claimed that the Saints were not going to allow the Giants to beat them deep.
"They weren't going to give us anything deep. They took away all the deep stuff and made us go underneath," he said.
"That was fine. We were going to just have to have some long drives and continue to play that way. There was a few times where they knocked us out of some good down and distance and we weren't able to sustain some drives."
Following Rosas' field goal, Kamara sandwiched the remaining two of his touchdown runs around a 1-yard touchdown leap by Giants running back Saquon Barkley.
Kamara's biggest run was his 49-yard scamper up the middle with 2:06 left in the game to make it 33-18.
"Yeah, he is special," Saints head coach Sean Payton said of Kamara.
"He works at it. I thought the offensive line and the defensive line were outstanding. When you can, like I said, finish a game like that on a power play, I'm proud of them. I'm proud of how these guys work. We had a good week of preparation."
The Giants' defense did all it could to stifle the potent Saints passing attack, but in the end, it managed just one sack, two quarterback hits and four passes defensed against Brees.
The Saints' defense mustered three sacks, four tackles for a loss, five quarterback hits and two passes defensed.
The game wasn't without its controversies, namely two instances involving the officials. The first came in the first half when Giants cornerback Janoris Jenkins was called for a horse-collar on a tackle against Kamara following a 6-yard reception in the second quarter.
That penalty put the ball at the Saints' 36 instead of their 21 and was instrumental in setting up their second score of the game.
The other controversy came on a non-call in the third quarter in which Shepard appeared to be horse-collared by Saints cornerback Ken Crawly.
The Giants have now failed to crack 20 points in three of their last four games and have only scored 73 points despite the return of Beckham, the addition of Saquon Barkley and the reported upgrade of the offensive line.
The Giants' immediate schedule doesn't get any easier as they visit the Carolina Panthers next week, a team coming off a bye, before returning home for a Thursday night game against the Philadelphia Eagles followed by another road game against the Atlanta Falcons.
Giants coach Pat Shurmur said the only thing they can do is work and coach themselves out of this hole they've dug for themselves.
"You just keep working," Shurmur said. "You just keep working and you play your way out of it and you coach your way out of it. Period. That's what you do, and that's the reality of it, and that's what I trust our guys and our coaches will do."