The Seadogs were seconds away from knocking the Gloucestershire side out in the first match at Flamingo Land Stadium until a last-gasp strike forced the tie to a replay last night, which Forest Green won 5-2 after an early blitz of goals.
Mr Bull said: “The problem was that we conceded two goals early on which killed it but it was a game of two halves.
"They dominated the first half and we more than matched them second half and played some very good football, with two super goals.
"I’m really proud of the fact that we’ve got this far and got a new record home crowd [in the home tie against Forest Green] so lots to be happy with.”
Athletic had only once been as far as the fourth qualifying round and never into the first round proper until this season.
Most of the players were experiencing this level of the famous competition for the first time too.
"Overall, I’m really proud of what we’ve achieved,” added Mr Bull.
"That was our seventh FA Cup game of the season last night – some teams will play fewer games and win it!”
Forest Green raced into a three-goal lead inside the first 30 minutes of the replay then swiftly added a fourth.
Boro fought back with goals from Alex Wiles and Michael Coulson a consolation.