Asher-Smith added the 4×100 metres title to her 100m and 200m crowns in Berlin on Sunday to become the first Briton to ever win all three at a major championships.
It wrapped up the final evening at the Olympic Stadium after Muir had taken the 1500m title – the first British woman to win the event – with the British men’s sprint relay squad also winning gold.
Asher-Smith anchored the team of Asha Philip, Imani Lansiquot and Bianca Williams to victory in 41.88 seconds, a world-leading time.
Asher-Smith, 22, said: “I came and won everything I entered so I’m very happy.
I’m happy, but it’s all about moving to Tokyo (the Olympics in 2020) and you’ve got to use it and push yourself on to a platform.
“The British athletes have had the Commonwealths and Europeans this year, so we’ve had to peak.
“Half the American people aren’t even running. While it’s great for me this year it doesn’t forecast how the next two years are going to go.
“When I got the baton I was like, ‘Not today, today is our day’ and I’m very happy to finish that off for the girls. I saw people and thought, ‘Not on my watch’.”
Muir became the first British woman to win the 1500m at the Europeans, adding the title to her 1500m and 3000m European indoor crowns she won last year.
GB's Muir and Weightman take historic gold and bronze in 1500m
The 25-year-old was overwhelming favourite and won in 4mins 02.32secs after a dominant display of front-running, with team-mate Laura Weightman claiming bronze.
Muir, whose gold was one of seven for the British team in Germany, claimed world indoor silver and bronze in the 1500m and 3000m respectively in March.
She said: “I’ve proved on a European level that I’m one of the best and one of the best in the world. But there are so many bright stars on a GB level.
Now it’s world’s outdoors (in Doha next year) and Olympics. That’s the next step. But I’m excited for the next few years.
“Me and Sifan (Hassan of Holland, who opted to race the 5,000m in Berlin) have gone back and forth so many times. I’ve won races. She’s won races. Now there is no clear favourite.
“Everyone is winning different races. It’s great to see. Going into any race, there are four or five girls who can win it. That’s exciting, but I’ve shown I can be one of the contenders.”
CJ Ujah, Zharnel Hughes, Adam Gemili and Harry Aikines-Aryeetey, a replacement for the injured Nethaneel Mitchell-Blake, stormed to victory in the men’s 4x100m relay to clinch the third British gold of the night.
GB men continue sprint dominance with 4x100m relay gold
Hughes added to the individual 100m old he claimed on Tuesday as the squad clocked 37.80s to win ahead of Turkey and Netherlands.
In the women’s 5,000m final Eilish McColgan claimed second after Israel’s Lonah Salpeter appeared to stop with a lap left.
McColgan takes silver for GB in dramatic 5,000m finish
Salpeter was challenging winner Hassan for the lead, but seemed to think the race was over at the bell and slowed down, allowing McColgan to motor past to win silver in 14:53.05. Salpeter came home fourth, only to later be disqualified.
Elsewhere, Swedish 18-year-old Armand Duplantis cleared a world junior record 6.05m to win pole vault gold.