Asher-Smith battled back from a hamstring injury to run the third leg of the 4×100 metres.
Dina Asher-Smith completed her Olympic comeback with 4×100 metres relay bronze in Tokyo.
Asha Philip and Daryll Neita add to the relay bronze they won in Rio while Imani-Lara Lansiquot claims her first Olympic medal.
More was expected of the squad but their baton changes let them down and they barely celebrated third.
Asher-Smith’s individual Olympic dreams were wrecked by a hamstring injury she suffered at the British trials in June.
She failed to reach the 100m final after running a time of 11.05secs – well below her 10.83s personal best – and then pulled out of the 200m.
The 25-year-old returned on Thursday to help the squad reach the final, qualifying fastest in a new national record of 41.55s.
Bronze added to Team GB’s athletics medal total after Laura Muir won silver in the 1500m earlier on Friday night following Holly Bradshaw’s pole vault bronze and Keely Hodgkinson’s 800m silver in the week.