Veteran Ronaldo tucked away a 39th-minute penalty to claim his 699th club goal.
The result left United third in Group E, behind Real Sociedad and early leaders Sheriff Tiraspol, who beat Omonia Nicosia 3-0.
The winners of the group stage, which begins from September 8, will advance directly to the round of 16 that will take place from February onwards.
United coach Eric Ten Haag is eager to bring in another goalkeeper to add pressure on first choice David de Gea.
Frankfurt are the first Bundesliga team to win the Europa League, or its predecessor the UEFA Cup, since 1997, when Schalke beat Inter Milan on penalties.
John Lundstram struck 10 minutes from time to give Rangers a 3-1 win over RB Leipzig and send them into the final 3-2 on aggregate after a see-saw clash at a rocking Ibrox.
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.
Real Betis defeated 3-2 Zenit St Petersburg after scoring twice in the first 20 minutes.