NEW ORLEANS (AP) — Rather than spend their week off dwelling on a dramatic Tom Brady pass that ruined their perfect record, the New Orleans Saints are taking a realistic approach to gauging their strong start to the 2013 season.

They were pleased to be 5-1, leading the NFC South and tied with Seattle atop the conference as their bye week arrived.

Locker room leaders, including quarterback Drew Brees and receiver Lance Moore, also were quick to point out the Saints could easily be 3-3, if not for last-minute successes early this season.

It took a fourth-down stand near their end zone to beat Atlanta in the season opener. The next week, New Orleans needed a last-minute drive — and a last-second field goal — to topple Tampa Bay in a game Brees said the Saints “maybe had no business winning.”

“So sometimes they go your way and sometimes they don’t,” Brees said. “You just hope you’re in a position where more of them go your way than not, because arguably half the games in this league are decided by the last possession, and at times the last play. You could say that certainly that’s been the case for our team. Three out of the six games have come down to literally the last play.”

The Saints have played their past three games without Moore, who expects to return when New Orleans plays against Buffalo in the Superdome on Oct. 27.

As much as it bothered the elder statesman of New Orleans’ receiving corps to watch from the sideline in New England, unable to help his offense run out the clock and keep Brady off the field, he wondered whether the loss might serve as an important motivator for a team that could have gotten too comfortable with itself had it gone into its short vacation at 6-0.

“It’s kind of a reality check, because a couple of those games could have gone the other direction, and I would say we’re extremely fortunate to be 5-1 and sitting in a pretty good position,” Moore said. “So I feel like we’re a good team, but I feel like we can definitely be a great team. We’ve got a lot of work to do to get there, a lot of things to sharpen up on and get some guys back, myself included.

“We’ll get healthier in this bye week, continue to put our foot on the gas with the way that we work, improving every day, and I think you’ll see a pretty good finished product.”

In any event, the Saints have come a long way in a year’s time. In 2012, with head coach Sean Payton suspended the whole season in connection with the NFL’s bounty investigation into the Saints defense’s cash-for-hits bounty system, New Orleans opened with four losses and never recovered, finishing 7-9 and out of the playoffs for the first time since 2008.

Payton returned last January, and players said it wasn’t long before his contagious competitive fire and knack for Chief Executive Officer-style quality control began to take hold again.

One of his first tasks was hiring defensive coordinator Rob Ryan to take over a unit which ranked last in the NFL in 2012, yielding an NFL-record 7,042 yards. Ryan’s group has been more opportunistic, coming out of Week 6 tied for sixth in takeaways with 12. New Orleans also did not give up as many as 20 points in a game until the loss to the Patriots.

Meanwhile, Payton has been calling plays again for an offense which was tied for fifth in yards per game (397.7) after six weeks.

Looking back at Payton’s more recent coaching record, perhaps New Orleans’ one-year turnabout in fortune shouldn’t be a surprise. Since 2009 — Payton’s fourth year as a head coach and the season New Orleans won its only Super Bowl — the Saints coach is 46-14, including 4-2 in the playoffs. That’s a winning percentage of .766.

“I’m encouraged that there are a handful of things that we’re going to be able to improve on and work on, and we haven’t played our best football and we’re sitting at 5-1,” Payton said. “You can easily look at a few games that could have gone a different direction, but they didn’t. I like the work ethic, the competitive nature of this team. I like it a lot.

” I think that getting back after this bye, there are a few things we’re going to be able to clean up and work to improve on as we get ready for this long stretch of the season.”

New Orleans Saints quarterback Drew Brees. (Associated Press)

Bill Feig