The Bundesliga side were in danger of being blown away when Benfica went 4-1 ahead shortly after halftime but a second away goal by substitute Goncalo Paciencia left the tie more finely balanced for next week's return.
Eintracht forward Luka Jovic, on loan from Benfica, missed two early chances as the Bundesliga side made a bright start before the game took a dramatic twist.
Joao Felix threaded a ball through the Eintracht defence to send Gedson Fernandes clear and he was pushed over from behind by Evan N'Dicka.
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The referee judged that Fernandes had been denied a clear scoring chance, N'Dicka was sent off and Joao Felix calmly converted the penalty in the 21st minute.
Jovic equalised against his parent club in the 40th minute after Benfica were caught in possession but Joao Felix put the hosts back in front three minutes later with a low strike from 20 metres which caught keeper Kevin Trapp off guard.
Ruben Dias headed a third for Benfica after a corner was knocked on in the 50th minute and Joao Felix sidefooted the fourth four minutes later as the Eagles threatened to run riot.
Eintracht were not finished, however, and substitute Paciencia, raised at Benfica's arch-rivals Porto, put them back in the tie with a looping header from a corner in the 72nd.
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