Swann, who is making her first appearance in a senior World Championships,and Glover came home in 7:15.13 minutes ahead of Romanian pair Roxana Cogianu and Nicoleta Albu to secure their progression.
Kathryn Twyman and Imogen Walsh laid down a marker to their rivals as they won their lightweight women's double skills, while brothers Richard and Peter Chambers held off the challenge of Greece to also reign supreme in the lightweight men's double sculls.
GB men's quadruple scull of Graeme Thomas, Sam Townsend, Charles Cousins and Peter Lambert came out on top in a tight battle with Estonia and Alan Campbell was another heat winner in the men's single scull.
There were second place finishes for the men's four – who produced a blistering last 500m sprint to take them from fifth to second - and the two lightweight single scullers, enough to see them all through to the next phases.
With six semi-final and two quarter-final places secured for GB there are eight more crews in action tomorrow, with Tom Aggar going in the para-rowing men's single scull and the men's and women's eights, the women's single and double sculls, the men's double scull and the lightweight men's four and pair all having their first races.
The men's pair and women's quadruple scull will have to contest repechages on Tuesday as they bid to remain in the competition having contested races where only the heat winner went onwards.