The quartet of Chris Brown, Demetrius Pinder, Michael Mathieu and Ramon Miller clocked a winning time of 2:56.72.
The USA finished second to claim silver, with Trinidad and Tobago in third.
Miller said afterwards: 'The US guys know we've been just behind them or thereabouts for a long time, but it's amazing we are finally taking home the gold.
'This is for everyone back home. We can't wait to celebrate.'