Arsenal were beaten 2-1 at St Mary's Stadium, with goals from Ross Stewart and Shea Charles sending Southampton into the semi-finals.
Norgaard, who played the full match, admitted the result was a major disappointment but stressed the importance of maintaining the right mindset.
He said: "The message is to have a positive body language, to talk with your teammates, with the coaching staff. Now is not the time to go with our heads down for too long.
"It's fine to be frustrated tonight and also tomorrow to analyse what went wrong, but then we also have to look forward because there are so many big games coming up for this club."
Arsenal had strong spells in the match but failed to take their chances. Norgaard pointed to key moments where the team lost control.
"Disappointing evening," he said. "We all had our hopes high to win today and go back to Wembley, but that's not the case. Sorry to disappoint so many travelling fans, they created an amazing atmosphere, and today was not good enough.
"It was a frustrating first half, but we had a really good period where we had them under pressure, created set pieces and shots around the box. The same in the second half before we conceded.
"We didn't manage to score in those periods and then we were too poor in defending the two goals."
Attention now turns to Europe, with Arsenal facing Sporting CP in the quarter-finals of the UEFA Champions League on Tuesday.