Trabzonspor, having drawn the first leg 3-3 at home, took the lead in Norway and looked to be going through right until the final minute of injury time - when Molde equalised to make it 1-1 on the night and 4-4 on aggregate.
With away goals now a thing of the past in UEFA competitions, the tie went to extra-time and, eventually, penalties.
But Trabzonspor won 4-3 on spot-kicks to progress to a showdown with Jose Mourinho's side.
The first leg of that tie will take place in Turkey next Thursday, with the second leg in Rome a week later.
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Turno di playoff: 19 e 26 agosto
Fase a gironi: 16 e 30 settembre, 21 ottobre, 4 e 25 novembre, 9 dicembre.
Sorteggio degli spareggi del turno a eliminazione diretta: 13 dicembre.
Spareggi del turno a eliminazione diretta: 17 e 24 febbraio.
Sorteggio degli ottavi di finale: 25 febbraio.
Ottavi di finale: 10 e 17 marzo.
Sorteggio di quarti di finale e semifinali: 18 marzo.
Quarti di finale: 7 e 14 aprile.
Semifinali: 28 aprile e 5 maggio.
Finale: 25 maggio.
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