Harriers big forward Drewe Broughton had the first crack at goal on four minutes, gaining possession near the halfway line and charging forward before driving wide from 30 yards.
The visitors were battling well early on and giving Cheltenham little to work with.
But there was a scare on eight minutes when keeper Clarke fumbled away MacDonald's shot from a low Goodridge cross.
Grayson returned the ball wickedly across the goal but Harriers managed to clear for the home side.
Then there was relief for the home side three minutes later when skipper Banks back headed Stamps's long throw only for the ball to land on top of his own net.
An exciting run from Cheltenham right-winger Goodridge came to nothing as the Gloucestershire side broke away swiftly.
Harriers took a shock lead in the 20th minute when Webb grabbed his first goal in the Football League.
It arrived when midfielder Bennett hooked a long hopeful pass into the area and keeper Book, under pressure from Hadley, could only palm the ball down for Webb to swoop. Dean Bennett
Within 2 minutes, the visitors could have easily made 2-0 when Doyle sent Broughton through the middle but the forward shot was too close to the advancing Book who made a good save to his right.
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