Mansfield ran the Gunners close, eventually losing 2-1 at Field Mill.
Clough said at the final whistle: "The effort they had was incredible, and we wanted to make it a proper cup tie and a tough game for Arsenal and make them uncomfortable, and I thought we did that.
"Overall, the lads were fantastic. We cut in from this angle a bit. You need a break in the box. We set the tone earlier on. We didn't want this 80% Arsenal having possession and controlling the game, or they would have picked us off.
"The number of efforts we have had today is absolutely brilliant."
On Mansfield goalscorer Will Evans, Clough added: "Very close to starting probably on Wednesday. He was starting, then we thought to go with Tyler (Roberts).
"I am extremely proud of everything and how close we got."