Struber said: "I am unhappy and disappointed with our performance. The spirit and energy from the opponent showed us what you need to do in an FA Cup game.
"Our spirit and energy, on the ball and off the ball, was not enough to win.
"We always need too long time to get in the right position. We need to come quicker in the right position where we have more space and more time.
"We lose so simple balls in both halves and this was not like a Championship team and we have to speak about this game with the boys."
Pompey dominated throughout and opened the scoring in the 37th minute when Andy Cannon teed-up Close to rifle the ball into the bottom corner.
Marquis doubled the lead on the stroke of half-time, poking Curtis' goal-bound header over the line.
Cauley Woodrow brought Barnsley back into contention with a 30-yard screamer just before the hour but their comeback hopes were short-lived when Curtis raced onto goalkeeper Alex Bass' long punt downfield to make it 3-1 three minutes later.
Christian Burgess sealed the win with a header from Steve Seddon's corner 14 minutes from time.
Barnsley striker Conor Chaplin marked his return to Fratton Park with a consolation goal in stoppage time against the club where he began his career.
Portsmouth manager Kenny Jackett said: "I felt there would be goals in the game. Barnsley, the way they set up, give people problems. They've given a lot of good sides problems recently, Championship defences.
"It's hard to win games of football, whether by the odd goal or four, that is great, for our players anyway.
"We want to do well in every competition, avoid injuries, that's a big one, hopefully build momentum and build confidence.
"I think we can be confident that we can build up a head of steam and have an exciting finale to the season.
"The cup has not necessarily been kind to us in recent years but as past winners it will be nice to go as far as we can and get some big games."