World silver medallists Lauren Bell, Sophie Capewell and Emma Finucane clocked 46.023 seconds to beat China.
Joshua Tarling, Rhys Britton, Will Tidball and Charlie Tanfield won their team pursuit final against Australia.
World champions Katie Archibald, Elinor Barker, Josie Knight and Anna Morris took women’s team pursuit silver.
There were also bronze medals for the men’s team sprint trio of Ed Lowe, Hamish Turnbull and Jack Carlin, and Jessica Roberts in the women’s elimination race.
The three-day event in Adelaide is the first of three Nations Cup rounds which will complete the qualifying period for the Paris 2024 Olympic Games.
Following the first round in Australia, riders will compete in Hong Kong from 15-17 March and Milton, Canada from 12-14 April.
Bell, Capewell and Finucane finished runners-up to Germany at last year’s World Championships in Glasgow and marked the beginning of their Olympic year with a comfortable final win against China, who suffered a mechanical problem.
“It feels incredible. I’m really proud of the girls and where we’re going,” Finucane said.
“This is only the start, and at the start of an Olympic year. I’m really excited for what’s to come.”
Tarling, Britton, Charlton and Tanfield finished strongly to cross the line in three minutes 48.469 seconds, more than one second ahead of Australia in the men’s team pursuit gold medal race.
But Archibald, Barker, Knight and Morris were made to settle for silver by New Zealand, who they beat to claim world gold last year.
The British quartet clocked trailed New Zealand’s Michaela Drummond, Ally Wollaston, Emily Shearman and Bryony Botha by 1.7 seconds at halfway in the final but improved to finish 0.349 seconds adrift in 4:10.578.
Lowe, Turnbull and Carlin finished in 42.915 seconds to edge China to team sprint bronze, as hosts Australia beat Japan to gold in 42.154 secs.
European bronze medallist Roberts earned a third-place finish in the women’s elimination race behind New Zealand winner Wollaston and American Jennifer Valente.