The interim boss said that he was delighted to give the fans a taste of the glory days, when Forest were a top-flight club competing with the likes of Arsenal.
"This has got to be something we build on," Brazil told BT Sport.
"This football club should be playing against the Arsenals of this world, and they should be beating them.
"These fans deserve to be watching Premier League teams and that's what we've got to drive towards."
Two goals from Eric Lichaj and penalties from Ben Brereton and Kieran Dowell helped Forest book their place in round four, with Arsenal's consolations coming through Per Mertesacker and Danny Welbeck.