GB had been victorious in the event at the past two Olympics but Germany proved the surprise package, dominating the race and setting a new world record of 4:04.242.
Archibald, Evans, and team-mates Laura Kenny and Josie Knight finished in 4:10.607.
Archibald told the BBC after the event: "We pulled it through to second best, a silver medal – three years to try again."
Debutant Evans added: "It's really special to be here. We're defending champions, there's big expectation for British cycling because we do have such a strong reputation, but there are so many strong nations that came out fighting.
"It wasn't to be this time, but we'll be back in Paris. ”
Kenny said afterwards: "Germany took everyone by surprise. We knew they were going to go fast, just not that fast."
The 29-year-old is already the most successful British female Olympian with four golds and one silver, while equestrian rider Charlotte Dujardin is the most decorated with three golds, one silver and two bronze medals.
The suspension of the Tokyo 2020 Olympics as a result of the coronavirus pandemic actually benefited the cyclist as she was able to recover from a broken shoulder sustained in January 2020.
Should Kenny win gold in the omnium or madison events later this week she will become the first British woman to win an Olympic title at three different Games.