Almgren finished in a time of 28:56.02, less than a second behind winner Jimmy Gressier of France (28:55.77), while Ethiopia's Yomif Kejelcha was second, who both ran season's bests to beat Almgren.
In hot and humid conditions in the Japanese capital, Almgren remained among the frontrunners through much of the race and hit the front with around 1000m to go.
But as the final lap began, the 30-year-old looked to be dropping back, as he was overtaken by Kejelcha and his compatriots Berihu Aregawi and Selemon Barega.
However, as Gressier powered through the field, Almgren pushed hard on the inside to surpass the fading Ethiopians to win Sweden the bronze.
Only 20th at the World Championships two years ago, Almgren was an agonising fourth in both the 2024 European and 2025 European Indoor Championships.
He is a man in form, though, setting a new European record in the 5000m in June, and now he has won his first major senior medal and put Sweden's name onto the medal table.