Papiss Cisse gave the Toon hope on 71 minutes after twice being denied by the assistant referee's flag, but Eduardo Salvio killed off the tie with a clinical finish late on
Trailing 3-1 from the first leg, the Toon knew they would need to mount a stirring comeback in order to make the last four of the competition, roared on by a passionate St James' Park crowd.
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The 4-2 aggregate defeat ends the European adventure which has proved a welcome distraction from a disappointing Premier League campaign, but Pardew is adamant his men could do no more.
"They are a good side - you can't let them get a lead against you," he told ITV.
"In the first half we kept it tight and looked to nick a goal - and almost did before half-time.
"We had two or three big chances at 1-0 but you can't ask for anything more from the crowd, the players.
"You've got to respect the opposition - technically they are a superb side.
"I am so proud of the performance - we couldn't have done any more.
"We've got some very good players - when we get them all back we've got a very good side."