Saints won the fifth round tie 1-0 thanks to a late penalty converted by Ross Stewart.
Eckert said afterwards: "Not so easy when you play a good first half and you feel you should be winning at half-time but aren't. Fulham have the quality and we had to accept the game where it was and in the end it was very well deserved.
"Nice ground to play. Quality is better than most other Championship stadiums, needed some moments in possession and we got them."
On disallowed goal in first half, he also said: "I have not seen that moment. Overall in the 90 minutes it's deserved that we go to the next round.
"The first two penalty takers were not on the pitch but we practiced some penalties midweek and Ross has taken penalties all over.
"If everything goes well we can play three games at Wembley this year. Just need to use this as game as fuel for the games coming up."