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It will be displayed on the men's and women's first team shirts from the 2022-23 season and will serve as a message of solidarity with more than 100 million people who have been forced to flee their homes.
The Catalan club will play next season's games at an almost full-capacity Camp Nou but will relocate to the Olympic Stadium in Montjuic for the following campaign.
While Barca were expecting huge backing at the stadium, ESPN reported there were close to 30,000 Eintracht fans among the 79,468 crowd.
Serbia international Kostic converted a fourth minute penalty and added another in the 67th with a low drive after Rafael Borre thundered in their second goal in the 36th for a 4-3 aggregate victory and a sensational win at the Camp Nou.
Atalanta goalkeeper Juan Musso saved a penalty and a follow-up attempt but he couldn't keep out an own-goal by Davide Zappacosta from the resulting corner as Leipzig salvaged a 1-1 draw.
Previous leaders Barcelona dropped to fourth in the 25th edition of the table, the Spanish club's lowest position since 2013-14.
A sold-out Santiago Bernabeu stadium watched silently as a much-improved Barca under the command of new coach Xavi Hernandez dominated from the start.
Real Betis defeated 3-2 Zenit St Petersburg after scoring twice in the first 20 minutes.
Benfica qualified as group runners-up with a win over bottom club Dynamo Kyiv.
Xavi probably could not have asked for much more in his first game managing a struggling Barca team who are sixth in the standings, eight points behind leaders Sevilla.