There was desperation in the cries, shrieks and rhythmic pounding that suffocated the silence out of the Mercedes-Benz Superdome for a few minutes Sunday afternoon.

Those watching knew the stakes. Maybe the season wasn’t coming to an early conclusion, but it was on life support. The offense’s two-point conversion attempt had just fallen short, and a Tampa Bay Buccaneers field goal on the ensuing possession would have put New Orleans down by eight.

Their team needed help. So the crowd, which had been knocked into a state of somnolence by a 24-0 Bucs run spanning the second and third quarters, came to life and played a pivotal role in helping New Orleans claim a 37-31 overtime victory. They got loud — perhaps as loud as they have been all season — and left Tampa Bay disoriented on a game-changing drive.

The unraveling happened immediately. After three penalties and a botched snap, the Bucs stood at the 1-yard line, facing a desperate defense being fueled by a desperate crowd. Running back Doug Martin was stuffed on the next play, and then Junior Galette busted through the line of scrimmage to drop Bucs quarterback Mike Glennon for a safety.

“The fans came out and really supported,” Galette said. “That really helped us. They got a false start — and that was a pre-snap penalty. A lot of credit to fans for being as loud as they are. We just came out and used that to our advantage.”

In the storybook version of this game, the reverberations from Glennon’s body hitting the turf would have knocked the Saints out of their funk. It didn’t happen that way. Drew Brees remained shaky down the stretch, and New Orleans (2-3) had to lean on its running game to put itself in a tie with the Atlanta Falcons for second in the NFC South.

But that’s fine with the Saints. After losing tight end Jimmy Graham (shoulder) and center Jonathan Goodwin (leg) to injuries that knocked them out of the game — if not longer — this team was simply happy to pick up its second win of the season and limp into its open date with an ugly win and their playoff hopes still alive.

“We wanted to have a win going into the bye week to put our hopes and spirits up,” running back Pierre Thomas said. “That’s what we did.”

This was the rare occasion when the running game lifted a passing attack that appeared hapless at times. Brees (35-of-57, 371 yards, two touchdowns, three interceptions) had little issue moving the ball on Tampa Bay before Graham was knocked out in the first quarter. Tampa Bay often used a safety to match up with Graham (two catches, 36 yards) — to the tune of 118 Saints passing yards in the opening period.

But with Graham out of the game, Tampa Bay often used a linebacker to cover New Orleans’ tight ends and did a better job taking away the other passing lanes. Brees began to struggle, and he shot himself in the foot by throwing a pair of interceptions that led to 14 Tampa Bay points — including one, after Brees was wrapped up by a pass rusher, that was run back 33 yards by linebacker Danny Lansanah for a touchdown.

“We scored 51 points today,” Saints tackle Zach Strief said. “Unfortunately, 14 were for the other team.”

Brees had another costly mistake that did not lead to points but exposed New Orleans to overtime. Following the Galette safety, the Saints tied the score on a 44-yard field goal by Shayne Graham and had a chance to put the game away after it got the ball back with 1:05 remaining.

New Orleans pushed to the Tampa Bay 49-yard line when Marques Colston dropped a pass over the middle and Brees narrowly missed tight end Benjamin Watson on another attempt. On third down, Brees spotted Robert Meachem streaking down the right sideline. His throw came up short and was easily intercepted by Alterraun Verner, sending the game to overtime.

The Saints put the ball in Brees’ hands to start the extra period, and he began by connecting with Colston for a 21-yard gain. He missed on his next three throws, but Tampa Bay’s Johnthan Banks, who was away from the action, was flagged for illegal hands to the face on third down, giving New Orleans new life and a new set of downs.

This time, the Saints approached things differently. After an 8-yard reception by tight end Josh Hill, New Orleans picked up 35 of its next 46 yards on the ground. Breaking down those numbers further, Khiry Robinson accounted for 33 of the 35 yards, including an 18-yard run to the left for the game-winning touchdown.

The running backs kept the Saints alive throughout the afternoon. Thomas seemed to show up throughout the game in, as coaches often refer to them, several have-to-have-it moments, and Travaris Cadet showed up with a few key receptions, including a 5-yard touchdown grab in the third quarter that made it 24-20.

And on the scoring drive earlier in the fourth quarter that pulled New Orleans within 31-26, Thomas broke loose around the right end for a 27-yard touchdown following a 12-yard gain by Robinson (21 carries for 89 yards).

It was also Thomas (four carries for 35 yards; eight receptions for 77 yards) who scored his first touchdown on a 15-yard reception as the blitz closed in on Brees and revitalized a screen game that has been mostly dormant this season.

“It’s been a while since we used them,” Thomas said. “(Coach Sean Payton) called it at the right time. It just hit for big yards.”

But there was no bigger moment in this game than the safety. There are several hypothetical scenarios surrounding it. One could ask whether New Orleans would have been better off if Tampa Bay punted, which would have given the Saints a shorter field and a better chance to score a touchdown.

The other hypothetical is that Brees would have thrown the same interception going for the end zone and New Orleans would be heading into its open date with a 1-4 record.

Given what actually happened, that play gave the Saints momentum and a reason to fight for life.

“We know how tough we are,” Galette said. “We have a team that wants to finish and guys with a lot of passion who want to finish.”

Now they can focus on finishing the season with that same passion.