The Nailers led 1-0 thanks to Danny Gordon, but the National League side eventually took a 2-1 victory with goals from Enzio Boldewijn and Kyle Wootton.
"The following of 1,700 at this level is remarkable and we hope this cup run has made a lot of memories for them and today [Saturday] adds to that for them to cherish," Black told the club's official website.
"Hopefully we can give them many more days out and we thank them for their support. It was incredible to see from the touchline."
On his side's performance, he added: "The lads make us proud every week.
"We said to them if they do everything they can to get a result out of the game and it doesn't happen then at least we can sleep well tonight.
"Everyone who has been to the game today [Saturday] will see the efforts the lads put in to take it so close against a team like this.
"We've got a level-headed group and we respected Notts County."