His arm strength declining, Brees does not scorch defenses with deep passes so much as bedevil them with shorter ones: According to the N.F.L.’s Next Gen Stats, his average throw has traveled just 6.3 yards, trailing only his backup, Teddy Bridgewater.

Once a catalyst for aerial domination, Brees, who turns 41 next month, has facilitated an offense more efficient than explosive, imposing in its precision. As ever, he leads the N.F.L. in completion percentage, and age has diminished neither his ability to manipulate defenses nor shape throws, such as his first touchdown, up the seam, against the zone, to Jared Cook.

Three seconds into the second quarter, Brees had tied a season high with three scoring passes. Four series into the game, Brees had thrown for more yards (172) than San Francisco’s defense had allowed on average (134.3).

To have a chance at beating the Saints, particularly amid the unforgiving din of the Superdome, Shanahan coached the game with the relentlessness of a jackhammer. On the first play after falling behind by 20-7, Garoppolo threw deep, tossing a 75-yard touchdown to Emmanuel Sanders.

Rather than fly back after losing at Baltimore, the 49ers spent last week practicing in Bradenton, Fla., at IMG Academy, where they installed a play that epitomized their approach: a reverse to Sanders. He executed it to perfection all week, until the last rep, and as Sanders readied to throw Sunday, he said, he thought to himself, “Man, don’t throw a duck again.”

Sanders lofted a 35-yard scoring pass to Raheem Mostert, who on the next drive helped propel San Francisco to a 28-27 halftime advantage with a 10-yard touchdown one play after taking a pitch — from fullback Kyle Juszczyk.

“We knew we had to take risks,” Shanahan said.

In the top-heavy N.F.C., that is how the elite distinguish themselves, with aggressiveness. The 49ers arrived at a fortress, lost several players to injury — among them center Weston Richburg (knee), carted off in the third quarter; defensive end Dee Ford (hamstring); and Sherman — and won the type of close game it did not last week, in Baltimore.