France took silver while the Americans recovered from a dreadful second baton exchange to pip Britain to bronze.
Double sprint champion Shelly-Ann Fraser-Pryce led the Jamaicans home for her third gold in a national and championship record time of 41.29, while Britain - who had been second at the start of the anchor leg - initially missed out on a medal as the inexperienced Hayley Jones faded badly in the home strait.
But France, who got silver, were disqualified and Britain got their bronze.
The Jamaica quartet of Carrie Russell, Kerron Stewart, Schillonie Calvert and Fraser-Pryce streaked to gold in 41.29, the second fastest time ever run.
A mishap on the second and third interchange between Alexandria Anderson and English Gardner who started her run too soon and had to halt to grasp the baton, effectively ended U.S. victory hopes.
France came second in 42.73 and it was only a storming final leg from Octavious Freeman that secured an unlikely American bronze, Freeman eating up ground down the home straight with a sizzling run to get into to a medal position.
The U.S. set a stunning world record of 40.82 at the London 2012 Olympic Games but none of that victorious quartet featured in the Moscow final, notably Allyson Felix who tore a hamstring in the 200m final and 100m bronze medal winner Carmelita Jeter who was left out.