Through 10 games, nearly every starter was available for a team that is in playoff contention for the first time since 2004.
During last week's loss to the Minnesota Vikings, the Rams suffered their first significant injuries in the starting lineup.
Leading receiver Robert Woods suffered a shoulder injury that will keep him sidelined for a couple of weeks, McVay says.
Cornerback Kayvon Webster entered concussion protocol and Nickell Robey-Coleman injured his thigh. Webster will be available Sunday, Robey-Coleman is listed as doubtful.
As they prepare to play the Saints, five players are listed on their injury report.
Rath and Scott are quick to acknowledge that luck plays a role in maintaining a healthy roster, but players say the Rams' good fortune has much to do with Rath's and Scott's communication and planning.
"They're player-friendly," cornerback Trumaine Johnson says. "If you go to them, they will listen."
Andrew Whitworth, a 12th-year pro, says he worked with "great people" during his 11 seasons with the Cincinnati Bengals, but that the difference between organizations is apparent.
"In Cincinnati unless you really went in the training room and asked for some help, or for somebody to be involved in something you do, there really wouldn't be any conversation with anybody about anything," Whitworth says. "So these guys, though, every single day they have a plan for each individual."
Scott has worked for the Rams for eight of his 15 seasons in the NFL. Rath, like McVay, is in his first season with the franchise.