Lancaster-born Swann, Katie Greves, Melanie Wilson, Frances Houghton, Jessica Eddie, Olivia Carnegie-Brown, Karen Bennett, Zoe Lee and cox Zoe de Toledo won the first ever British medal in the event having come through the field late on.
Reigning champions the United States were deserved winners of a race that Britain started slowly, sitting at the back of the pack at the halfway mark.
However, they upped the ante and entered the final 500 metres in third, before kicking on and holding off Romania to win silver.
Swann, a former world champion in the pair, said: “To be honest, with 500m to go I thought we were going to win.
“I just absolutely believed in every single stroke that we had what it took.
“The silver medal is incredible – these girls are strong, they’re sassy, they’re inspirational and I’m just so happy to row with them.”