The Magpies took an early lead through Papiss Cisse's strike but were pegged back thanks to strikes from Rodrigo, Lima and Oscar Cardozo's penalty, given for a Steven Taylor handball.
"I thought we did so much right tonight," Pardew told reporters after the game. "We got ourselves in a great position and, especially at 1-1 in the second half, we had kind of taken the sting out of them.
"And then we gifted them a goal and then, looking at the third goal, I think Steven's desperately unlucky. It looks like he's caught Moussa [Sissoko]'s boot and that's affected his jump so it just wasn't to be I'm afraid.
"If it had been 2-1 you'd like to think we'd have a fantastic chance at home but it's an uphill chance for us now.
"It's one of those things and as a group we can be proud of the performance but disappointed in the result. I think that probably sums it up."
Despite a two-goal deficit, Pardew is confident that a roaring St James's Park atmosphere can inspire Newcastle to progress to the semi-finals when the two sides meet again on April 11.
"I think the away goal could come into it," he suggested. "If we get the first goal, I think that will obviously be very, very important at St James's.
"If we get the first goal, we're only going to need one more but they are a quality side and we have got to give them respect. On the break, they are going to be dangerous.
"I think we showed signs tonight that if we're ambitious, we can hurt a team."