Dan Bigham, Ethan Hayter, Ollie Wood and Ethan Vernon helped Britain win their first title in the discipline since 2018, clocking 3:45.829 to finish 0.2 seconds before defending champions and world record holders Italy.
Denmark took the bronze medal after defeating Australia for third place, snatching the final podium spot by 1.4 seconds.
In the women's event, the Italian quartet of Elisa Balsamo, Chiara Consonni, Martina Fidanza and Vittoria Guazzini finished in 4:09.760, ahead of Britain's 4:11.369, to take gold.
France delighted the home crowd by beating Australia to bronze.
Dutch sprinter Harrie Lavreysen, who won individual and team golds at last year's Tokyo Games, won his third straight keirin world title in a time of 9.751, beating compatriot Jeffrey Hoogland and Colombian Kevin Quintero to the top of the podium.
Belgium's Lotte Kopecky improved on last year's runner-up finish in the women's elimination race by bagging gold, with Italy's Rachele Barbieri finishing second and Jennifer Valente of the United States taking bronze.
Canadian Dylan Bibic, 19, held off Japan's Kazushige Kuboki for a famous victory in the men's 15km scratch race. Roy Eefting of the Netherlands completed the podium.
