Hereford started brightly and almost took the lead in the second minute as Marc Pugh's corner found Godsmark at the backpost but the unmarked Newcastle loanee sent his volley over the bar.
Charlton – making eight changes to the side that beat Wycombe on Saturday - started to get a foothold in the game with a quarter of an hour gone and Matthew Spring teed up Andy Gray but the striker headed tamely over.
Pugh continued to look lively for Hereford striking the post in the 22nd minute after a mazy run that saw him leave two Charlton defenders stranded.
Former MK Dons frontman Izale McLeod was the next to go close as he looped a header just over for Charlton after half an hour, while Pugh released Godsmark and in turn Glen Southam, but his shot was charged down by the visitors' defence as the half ended goalless.
The game looked to have turned in Charlton's favour just after the hour as the visitors were awarded a penalty when Richard Rose was adjudged to have dragged Gray to the deck.
But after dusting himself down to take the spot kick Gray's powerfully struck penalty was well saved by Bulls stopper Adam Bartlett.
Prior to the penalty the hosts had continue to look the more dangerous with the ever-dangerous Pugh again going close, while Rose saw a header drift just over.
There were few chances in the remainder of the half as the game drifted towards extra time and it was there in the 98th minute where the breakthrough finally came.
Darren Dennehy was the architect for Hereford carrying the ball out of defence before finding Matt Done whose cross was met by 19-year-old Godsmark who crashed the ball home from close range.
Charlton threw everything at Hereford in the second half of extra time with Stuart Fleetwood, Deon Burton and Gray all wasting good chances as the Bulls clung on.