James Dasaolu, Adam Gemili, James Ellington and CJ Ujah took the men's 4x100m title in 38.17 seconds.
Emily Diamond, Anyika Onuora, Eilidh Doyle and Seren Bundy-Davies won the women's 4x400m title in three minutes 25.05 seconds.
Britain claimed 16 medals - including five golds - their highest tally at a European Championships in 58 years.
Poland (12 medals, six golds) finished top of the medal table, with Germany (16 medals, five golds) second, ahead of Great Britain.
Britain were also second in the women's 4x100m relay, and third in the men's 4x400m final.
Dasaolu, Gemili, Ellington and Ujah defended Britain's men's 4x100m title, won in Switzerland two years ago, by finishing 0.21secs ahead of France in second, with Germany third.
Ellington, along with Gemili, was part of the men's team who won 4x100m gold in Zurich two years ago.
"Everyone is running fast in our team, so the only difficulty is to select the right team because we have so many good runners right now," he said.
"We could have had a number of different team compositions today. Yet any of our teams would have done great."
The women's 4x400m team set the fastest time in the world this year as they won Britain's first European gold in the women's 4x400m since 1969, pushing France into second place, while Italy took bronze.
Onuora, who won an individual 400m bronze on Friday, said: "We've got a great team. It was a perfect finish. It couldn't have been better."
Asha Philip, Dina Asher-Smith, Bianca Williams and Daryll Neita secured 4x100m silver by finishing just ahead of Germany while Dafne Schippers' Netherlands team were clear winners.
