Newcastle will play Bedford in the Championship final after a 15-6 semi-final second-leg victory over Leeds that gave them a 34-30 aggregate win.
Newcastle face Bedford in a two-legged final although, as it stands, only the Falcons can win promotion because the Blues' Goldington Road ground does not meet Premiership criteria.
Newcastle lost the first leg of their semi-final after conceding a last-gasp try to Leeds number eight Ryan Burrows.
But Gopperth wasted little time in erasing that deficit, landing four penalties to put Newcastle 12-0 ahead before Leeds fly-half Joe Ford kicked the visitors onto the board.
The Falcons dominated the second half and could have closed the game out but Gopperth, having landed his fifth penalty, then missed two shots at goal to keep the door ajar for Leeds.
Ford did land a drop-goal but Newcastle, who had finished top of the table at the end of the regular season, saw out the victory and sealed their place in the final.