Richardson turned on the afterburners to overhaul Britain, Germany and France in the home straight as the US quartet took gold in 41.78sec.
Britain took silver in 41.85sec while Germany finished with bronze after crossing in 41.97sec, just edging out host nation France.
The victory represented a redemption of sorts for Richardson, who had been heavily favoured to win the individual 100m crown earlier in the Games only to be upset by St. Lucia's Julien Alfred.
In driving rain at the Stade de France, the US had looked to be struggling to get into the medal positions after Melissa Jefferson, Twanisha Terry and 200m gold medallist Gabby Thomas combined to leave Richardson in fourth place as she collected the baton on the last leg.
But the fiery 24-year-old Texan would not be denied, accelerating through the gears to sweep past Britain's Daryll Neita, Germany's Rebekka Haase and France's Chloe Galet.
Richardson could even afford a cheeky glance to her right as she passed her rivals before taking the tape.
"I just remembered trusting my third leg, trusting Gabby, knowing that she's going to put this thing in my hand and to leave my best on the track," Richardson said after her stirring last leg.
"It feels amazing," US second leg runner Terry added. "Each and every one of us came out to these games to strive for greatness, and I think we did a very good job of that whether it was individually or together."
