The 200m world champion ran the third leg as the British women clocked 41.88 seconds in Tokyo on Friday to finish behind Jamaica and the USA.
Asha Philip and Daryll Neita added to the relay bronze they won in Rio while Imani-Lara Lansiquot claimed her first Olympic medal while the men won silver after being beaten to gold by Italy by 0.01 seconds.
Asher-Smith’s individual Olympic dreams were wrecked by the serious hamstring injury she suffered at the British trials in June but she remains grateful she made it to Japan.
“I’ve got a bit of a mixed-emotions feeling because obviously I’m like, ‘aagh, I could have done this and this and we could have pushed for a different colour medal’,” she said.
“But at the same time if I’m being honest, on a personal journey, I was on crutches six weeks ago and there was a less than 10 per cent chance I was going to be here.
“I worked incredibly hard and to come away with a medal here is honestly something I could not even have contemplated six weeks ago. Seven weeks ago? Absolutely. Six weeks ago? Not at all.”
Asher-Smith 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.
She added: “I knew that what actually hurt so much about the 100m and 200m is that all I need is another week, two weeks and then I’ll be back to my normal speed.
“I’m obviously moving faster here and yesterday than I was last week but that’s obviously how recovery from these things goes.
“You just improve every day so I knew we had another week until the relay... train, train, train and make sure I was as fast as possible in the time that I had for these ladies.”
Neita did reach the 100m final, coming eighth, and also made up the ground needed in the final leg after taking the baton with Great Britain fifth.
“I haven’t actually seen it (the final leg). I was just patiently waiting to see Dina and she came flying into me as she would,” she said.
“I just did my best to bring it home for the girls and honestly we’ve worked so hard as a team, this medal means a lot to us, we’ve been through so much.
“I’m just super happy, it’s been an amazing championships for me personally: first sub 11, made the Olympic final and now coming home with a bronze medal, I can’t really complain.”
The British men’s 4x100m team claimed silver after being beaten on the line by Italy. CJ Ujah, Zharnel Hughes, Richard Kilty and Nethaneel Mitchell-Blake ran 37.51 seconds to be beaten to gold by 0.01 seconds.
It completed a remarkable summer for Italy after they won Euro 2020 and saw Marcell Jacobs take 100m gold in the Olympic Stadium on Sunday but Kilty sent out a warning ahead of next year’s World Championships in Eugene.
He said: “In the history of mankind this is Italy’s biggest year ever. Forget the Roman Empire, Nero, all that nonsense. This is as big as it gets for them.
“They better soak it in because they have conquered the world this year.
“What more can we say, we lost by a hundredth but God was looking over Italy. I don’t know where they dragged that from. We’re better than them and we’ll beat them next year.
“When I was 10 and if someone said you will win an Olympic silver in Tokyo, I would have snatched your hand off. When I am 50 and fat and looking back on my career I will say that is one of the best nights of my life.”
It was bittersweet for Hughes after he was disqualified from the 100m final for a false start.
He said: “Tonight is a night where it’s special for me and the team, what happened in the past is in the past. I needed to gather myself and I did that.