Archibald, Laura Kenny, Neah Evans and Ellie Dickinson defended the title they won in Glasgow last year by beating Germany in the final.
The margin of victory was almost three seconds as the GB team recorded an impressive time of 4:13.828.
Nelson, winner of the scratch race on Wednesday, also took part in the elimination race, comfortably keeping herself out of trouble as the field was whittled down.
After seeing off Polish bronze medallist Nikol Plosaj, she found herself in a contest for gold with Dutch legend Kirsten Wild, and she pushed the home favourite all the way before Wild prevailed by a narrow margin as the two battled on the final lap.
Archibald is Britain’s representative in the women’s individual pursuit today.
The men’s team pursuit squad, Ollie Wood, Charlie Tanfield, Ed Clancy and Ethan Hayter, beat Switzerland to win bronze.
In the men’s sprint, Scotland’s Jack Carlin – at the velodrome where he won world championship silver last year – was beaten by Poland’s Mateusz Rudyk in the quarter-finals.