Safety Jairus Byrd was not with the Saints last year when they finished fourth in the NFL in yards allowed under new defensive coordinator Rob Ryan.

From watching film, though, he knows one thing for certain. This season’s defense, which ranks fourth-to-last in yards allowed after four games, is not recognizable compared to the group from 2013. Byrd was supposed to be the final piece to the puzzle. Instead, the entire defense has been one big puzzle.

“I’m definitely surprised with the things I’ve seen from our defense in the first quarter of this year compared to how it was last year,” he said. “We have to take away something. Something has to be shut down to make teams one-dimensional.”

Outside linebacker Junior Galette was a key part in the Saints’ defensive success last season, registering 12 sacks while the team had 49. They have just five sacks through four games this year, a statistic he attributed to their struggles against the run.

DeMarco Murray ran for 149 yards and two touchdowns on 24 carries in Dallas’ 38-17 victory on Sunday night.

“Our strength is we have guys that can get to the quarterback,” Galette said.

“If we don’t stop the run, there’s no way possible that we’re ever going to get a chance to get to the quarterback, because he’s going to run the play-actions and just dump it off as soon as we try to get there.”

Fake failure

Upon further review, Sean Payton said the Saints should have punted midway through the fourth quarter instead of trying a fake punt that fooled no one.

On fourth-and-9 from the New Orleans 41, gunner Travaris Cadet went in motion to the left and ran right in front of punter Thomas Morstead on a misdirection play that had Morstead field the snap and roll to his right.

Receiver Josh Hill was covered tightly. Morstead, whose only pass during his NFL career came during a Pro Bowl, could not find anyone open and was tackled for a 2-yard loss.

“That’s on me for being impatient,” Payton said. “If I had it to do over again, we would have punted and tried to come up with another stop and worked to cut into that lead. Thomas did exactly what he was supposed to.”

Morstead said he never considered throwing and hoping Hill made a play.

“If there were three or four minutes left, I probably would have taken the chance,” he said. “With 71/2 minutes left, we still had the chance to get the ball back. Everybody faked left, so if I throw a pick, there were probably five (Cowboys) on this side of our team, so it would have been a touchdown.”

Drops party

Tight end Jimmy Graham dropped Drew Brees’ first pass of the game, setting the tone, even though the Saints benefitted from a defensive holding call.

Graham fumbled after a reception in the second half. Marques Colston dropped a pass that was close to a fumble.

And Kenny Stills dropped a pass near the sideline that would have given the Saints a first down at the Dallas 47 when they trailed 31-17 in the fourth quarter. The ill-fated fake punt came a few plays later.

“That (the dropped passes) was one of the things that obviously we made notice of (in watching video of the game),” Payton said. “Obviously that hurt us yesterday. They stalled some drives.”

No passion?

Byrd understands the criticism of the defense after its subpar play, but he rejects the notion the Saints do not care enough.

Of the unit’s many flaws in September, he insisted a lack of passion was not one of them.

“No one wants to win more than we do,” he said. “We’re the ones putting in the time and effort in practice and watching film. We’re passionate about it.”

Lagniappe

Khiry Robinson’s 62-yard run in the fourth quarter was the Saints’ longest since Deuce McAlister had a 71-yarder run against Carolina in the 2004 season finale. ... Opponents have run 263 plays since the Saints’ lone takeaway, a fumble Byrd forced in the first quarter of the opener at Atlanta. Every other team in the NFC has forced at least three turnovers.