The British quartet of Dina Asher-Smith, Imani Lansiquot, Amy Hunt and Daryll Neita secured women’s 4×100 metres silver at the Stade de France.
Team GB won bronze in the same event for the last two Olympics and were one of the favourites on Friday night.
They looked like they might be on course to secure gold before a messy handover between Hunt and anchor Neita cost them time.
Silver is Great Britain’s best result in the Olympic women’s 4x100m relay in 68 years.
Asher-Smith said: “We are all so proud and so happy. We have worked so hard for this, so to be able to be in a position not just to challenge for a medal, but also for gold, it’s phenomenal.
“I tried to get to Imani first and I wanted to set up the ladies for a phenomenal run. I did not want anybody else in sight.
“I love how the championships are giving medals and the full squad to have their moment. This is a team effort.
“We have worked so hard as a women’s team relay squad since 2012, we have pushed so hard for the last ten years. To stand on the podium with everyone is a very beautiful moment.”
Neita added: “No frustrations at all. It’s a relay and anything can happen and in amongst all the chaos we remained professional, got the job done and got the baton round.
“That’s what it’s all about because it’s a relay and anything can happen, it’s really not guaranteed. We came in in a really strong position and we left with a medal and I think we should be really proud of ourselves.
“We are coming home with a silver medal at the Olympic Games, we’ve upgraded our two bronzes to silver and it was with the help of the other girls as well, we’ve got Desiree (Henry) and Bianca (Williams) who ran in the heats and they’re great.
“We’ve got our coaches that get us into great shape and all in all just Team GB and the British public just supporting us and always giving us that backing.
“This is really special and we’re going on the podium, this is insane. And the boys, we’re bringing two medals home so hopefully you’re just proud of us.”
The British men followed the women’s silver immediately with 4×100 metres bronze.
It was a season’s best for the British men’s quartet in 37.61, who finished behind Canada, champions in 37.50, and South Africa, who claimed silver.
The United States’ team were disqualified for passing the baton outside the takeover zone in the first exchange between Christian Coleman and Kenny Bednarek.
Azu said: “It’s unreal, for me the champs didn’t start great (after a false start in the 100m) so to leave with something, we wouldn’t be in this position without each other, we trust each other. The boys put their trust in me to start us off.”
Hinchliffe added: “You could see it was a really messy race but we gave all of it. It was all heart, all pure speed and we gave 110 per cent. You could see how much we wanted it and we got the result so its great.”
The 23-year-old had already collected bronze with the mixed 4×400 metres relay team in Paris, but her individual time landed her fifth in the final.
Dominican Marileidy Paulino stormed to gold in an Olympic record 48.17 seconds, Salwa Eid Naser claimed silver for Bahrain, and Natalia Kaczmarek rounded out the podium.
Ireland’s Rhasidat Adeleke narrowly missed out, finishing fourth in 49.28.
Britain’s Eilish McColgan finished in 15th position in the 10,000m in a season’s best 31:20.51, with her teammate Megan Keith 23rd in 33:19.92. Kenya’s Beatrice Chebet took gold.