The New Orleans Saints looked like a leaking boat.

The injuries mounted even before the game began. Safety Jairus Byrd made it onto the cover of Sunday’s game program, but he never made it to the field after season-ending knee surgery. Once in play, tight end Jimmy Graham, linebacker Ramon Humber, center Jonathan Goodwin and cornerback Patrick Robinson formed a painful parade to the bench.

Then the lead slipped away. A 13-0 edge midway through the second quarter over the Tampa Bay Buccaneers dissolved into a 24-13 deficit by the third in the span of about 5½ minutes of game time — fueled by two very forced Drew Brees interceptions, one a pick six.

Those fans who showed up at the Mercedes-Benz Superdome were booing and/or leaving. It appeared the Saints’ season of high expectations was spiraling down the drain.

The Saints looked like they were in desperate, perhaps unrecoverable straits. Between the Sean Payton commercials floating across the video replay boards, telling us football is a tough game (no kidding, Coach), some song lyrics came to mind:

Save my life

I’m going down

For the last time

Somehow, the Saints didn’t sink. They found their depth. Pushed to the brink, they pushed back, even after the Bucs took a 31-20 lead with 13½ minutes left. They roared in response with 17 unanswered points, capped by Khiry Robinson freight-training Bucs safety Bradley McDougald (what was with that train sound effect in the Dome on Sunday?) on his way to scoring an 18-yard, game-winning touchdown for a 37-31 overtime victory.

It was just one win and it was just the Bucs, who are now where they were widely expected to be, lurking at the bottom of the NFC South fishbowl.

Whether the Saints will be battling for a division title or playoff berth when they return the trip to Tampa in the Dec. 28 regular-season finale is a matter of enough conjecture that it can easily occupy everyone’s time between now and New Orleans’ next game, Oct. 19 at Detroit.

But at least the Saints are still in the hunt. They’re 2-3, only a game behind division-leading Carolina. The engine sounds choppy and they’re trailing an oil slick, but the battered beast is still afloat.

“We didn’t give up,” said running back Pierre Thomas, a big part of the Saints’ rushing attack that was a big part of their victory. “There were times we took a step back, but we never gave up hope. We fought and fought and fought.”

There were times when the Saints were fighting themselves, their own demons, as much as the Bucs. One immediately thinks of Brees’ first two interceptions — the first setting up Tampa Bay’s first touchdown, the second an outright pick six by linebacker Danny Lansanah for that 24-13 lead with 11:20 left in the third.

Both plays were times when Brees’ protection broke down and he should have gone down with the ship, um, ball tucked under his arm. But a 1-3 start to a season tanked up with about 120 PSI of Super Bowl contender helium had Brees trying to do too much.

“The first two shouldn’t have happened,” said Brees, who also threw a comparatively harmless interception near the end of regulation intended for Robert Meachem, a risk at the time worth taking. “They gave them 14 points and shifted momentum when we felt the game was firmly in our grasp.”

When the Saints fell behind by 11, they could have easily run one of Sunday’s white jerseys up a goalpost and set the snooze alarm for 2015.

But they didn’t. Showing a remarkable amount of resolve and patience given the increasingly gruesome circumstances, the Saints went back to what was working well earlier in the game — judiciously mixing the pass with a running game that remains a bedrock for this offense even without the valuable Mark Ingram (did I really just write that?) the past three weeks.

The defense made some clutch stops, including that huge sack by Junior Galette on a questionable Bucs decision to let Mike Glennon pass from his end zone up 31-26. Then in overtime, the Saints offense simply overpowered the wilting Bucs defense. After five straight passes to start overtime, the Saints ran on six of the final eight plays, including Robinson’s touchdown.

“Especially in that closing drive, we had some big runs that were well-blocked and gave us an opportunity to get a little more than maybe we were expecting,” Payton said. “Get another first down and not have to kick a 34- or 45-yard field goal, but get ourselves closer and closer. It was great to finish with a run like that.”

Now the Saints can rest up, heal up and work on home improvement through their open date.

Going down for the last time? Not this time. The Saints still have hope. After the upheaval of the first five weeks, that’s definitely worth something.

Follow Scott Rabalais on Twitter @RabalaisAdv.