Elvis Hammond put the Robins on front on the hour with a low shot before Godsmark levelled for the Bulls with 14 minutes remaining.
There were few chances for either side in the first half.
Hereford's best effort saw Lee Morris head a Godsmark cross over the bar, while Oliver Bozanic's 20-yard shot forced home goalkeeper Adam Bartlett into a full-length save.
Five minutes into the second period Justin Richards should have put the Robins in front when he met a cross from Bozanic but his close-range header dropped wide.
However, Cheltenham went in front in the 60th minute when Hammond escaped his marker before beating Bartlett from eight yards.
But Hereford hit back as Godsmark latched onto Kenny Lunt's pass before bursting into the area and slotting the ball past Scott Brown.