The Scot ran three minutes 54.50 - a new British record - in Tokyo on Friday night as Kenya’s Faith Kipyegon claimed gold, defending her title, with Sifan Hassan winning bronze.
It ended Hassan’s remarkable triple bid on the track having already won the 5000m. She will also run in the 10000m on Saturday after missing out on an unprecedented hat-trick of wins.
The 28-year-old hit the front inside the first lap with Kipyegon and Muir responding along with Canada’s Gabriela DeBues-Stafford.
But Kipyegon took the lead early in the final lap and Muir moved into second with 200m to go.
Muir’s medal in Japan ends a run of several near misses at major championships.
She came seventh in the 1500m in Rio after fading despite being in medal contention with 200m to go; the disappointment followed the 2017 World Championships where she finished fourth by 0.07 seconds.
Since then she has become a quadruple European indoor champion, European champion in 2018 and claimed silver at the World Indoor Championships.
Muir also came fifth at the World Championships in Doha in 2019 after seeing her preparation hampered by an Achilles problem but finally reached the podium at the Olympic Stadium.
Earlier, the men’s 4x400m relay squad failed to make the final after coming sixth in their heat.