Hangeland headed home against the run of play in the first-half and Fulham, 2-1 up from the first leg, made it through despite Jadson’s strike for Shakhtar Donetsk levelling the game on the night.
Shakhtar, the 2009 UEFA Cup winners, were sent spinning out as Fulham’s European adventure took them further than they have been in their history in the revamped competition.
Hodgson said: “It has been a wonderful night for the club.
“I don’t like to compare nights, but it’s yet another marvellous night along the way and great credit to the team.
“I really enjoyed the night and the performance but I still hope there’ll be some more good nights to come.’’
Having stood up to massive Shakhtar pressure, incredibly it was Fulham who took the lead with their first attack of the night. Damien Duff floated in a cross and Hangeland’s glancing header beat Andriy Pyatov.
The moment Fulham feared came in the 69th minute when Douglas reached the byline and pulled the ball back for Jadson to sweep past Schwarzer to level the match on the night.
But they survived the final 20 minutes and the stoppage-time dismissal of Danny Murphy, for kicking out at Darijo Srna, to advance.
SHAKHTAR DONETSK: Pyatov, Hubschmann (Kravchenko 46), Kucher, Rat, Rakytskyy, Fernandinho, Jadson, Ilsinho, Willian (Douglas Costa, 54), Srna, Luiz Adriano (Gladkiy, 76).
FULHAM: Schwarzer, Kelly, Hangeland, Baird, Hughes, Gera, Murphy, Duff, Etuhu, Davies (Riise, 89), Zamora (Elm, 72).
Referee: Svein Oddvar Moen (Norway)