The Ducks were beaten 3-1 at home to Mansfield on Saturday lunchtime in front of the BT Sport cameras.
Over 1,700 were in attendance at Sheerien Park for the lunchtime tussle.
Elliott's side, who ply their trade three divisions below the Stags, put up a brave fight drawing level in the first half, hitting the post and then seeing goalkeeper Jon Stewart save a penalty.
But they couldn't prevent the visitors' Danny Rose, a Barnsley lad, from notching twice in the last 20 minutes as they exited the world's oldest cup competition.
Speaking to the Yorkshire Post, Elliott said: "We wanted to be competitive and show them what we are all about.
"I thought that we had them on the ropes for a little while and it's taken an incredible finish for the third goal to kill the game off.
"When Jon saved the penalty, you start thinking it's our day.
"But, fair play to the lad (Danny Rose), I think he's a local lad and ex-Barnsley.
"They were two incredible finishes.
"Especially the first one, because I thought it was a great leap and a great header."
Elliott admits that his side ultimately missed their chance to seal a famous victory when the scores were even at the break.
"We hit the post and came into the game slowly," said Elliott.
"I'm sure at half-time Mansfield would have been slightly worried.
"It was nice to be in that position, it's just a shame we just couldn't turn the screw in the second half.
"Fair play, they deserved to get the win."