John Melligan's 52nd-minute corner was not cleared properly, and Kayode Odejayi fired home.
The FA Cup fourth round tie at Cheltenham Town will be beamed live on BBC television, with a Saturday 12.30 kick-off.
Western Express on Wednesday: "Cheltenham Town will be able to add to their squad after hitting the FA Cup jackpot. A 1-0 third-round replay win at Chester last night means Newcastle United and the Match of the Day cameras will visit Whaddon Road a week on Saturday.
"Chairman Paul Baker conservatively estimates that the Robins will profit to the tune of £225,000 from TV and radio cash plus £40,000 winner's fee and gate receipts."
Goal hero Kayode Odejayi summed up the mood of the players: "They don't come much bigger than Newcastle so it will be a great day for the club. Everyone is delighted. The team's performance was magnificent."