Stef Collins' team battled hard at the showpiece in Paris after securing qualification in Romania earlier this summer.
GB went down 15-12 to Russia in their opening game with the Olympic silver medalists pulling away in the final seconds.
Then, in the shadow of the Eiffel Tower, they were edged 20-18 by the Ukraine to miss out on the opportunity of getting out of the group stage.
The team, made up of Temi Fagbenle, Shanice Beckford-Norton, Shequila Joseph and Cheridene Green, grafted hard and will move on to prepare as Team England for the 2022 Commonwealth Games.
“I’m super proud of the way we competed in our first Europe Cup Finals. We stepped into unknown territory and can’t fault our effort as we had opportunities to see us advance through the pool stage. We’ll keep learning and the most important takeaway - the more we play this format the more success you’ll see from our GB squads. We value every opportunity we get to step on the court and represent GB. This is only the beginning.
“And credit where credit is due, we may be new kids on the block but you can see how so many nations have embraced 3x3 and it certainly shows with the way they play.
“We’re all looking forward to the next opportunity and will use this experience to further our preparation for the 2022 Commonwealth Games in Birmingham.”
Serbia's men and Spain's women won the respective competitions. Serbia won their third consecutive 3x3 Europe Cup title in the men's category after defeating Lithuania 20-14 in the final.
The Tokyo Olympics bronze medalists dominated the title game against Lithuania thanks to Miroslav Pasajlic and Mihailo Vasic, who scored 8 points apiece.
In the women's competition, Spain avenged their overtime final loss in the last edition after beating first-time medalists Germany in the final (16-12).
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