Steven Naismith put the Toffees in front at Craven Cottage on Tuesday night but second-half goals from Dimitar Berbatov and Darren Bent booked Martin Jol's side a fourth-round place.
Distin told the Liverpool Echo: "We played really well in the first half and had the game in our hands. Realistically we should have been 2-0 up by half-time. We created a lot of chances but could only get the one goal.
"I don't believe half-time came at the wrong time for us, you always know when it's coming and it was up to us to carry on in the same way in the second.
"We felt comfortable but in the second half they changed their system a bit, started to press higher and we didn't adapt.
"It was a completely different game after we conceded their first goal."
Everton were strolling to victory after dominating the first half, but Distin felt poor defending robbed his side of a place in the next round.
"The second goal was just about communication but we needed a little more focus as it was an easy set-piece. It was an easy pass and we didn't switch on. It wasn't good enough," he added.
"It's difficult to think of any positives at the moment. We're just feeling disappointment right now. I feel like it was a game we could and should have won."