The visitors broke the deadlock in the 10th minute when Sam Deering jinked his way into the box and pulled the ball back for Byron Harrison to apply a cool finish.
Earlier, Northampton had gone close when Norris fired over from the edge of the box, but Harrison almost grabbed his second when he headed wide of the far post from Troy Brown's cross.
Cheltenham continued to dominate as Northampton toiled without success to get back on level terms.
That was until just before the break, though, when Keith Lowe and David Noble got in a tangle and let in Norris, who rounded keeper Scott Brown before firing home from a narrow angle.
The goal didn't stop Cheltenham from getting back on top after the restart and Harrison missed a great chance to restore their advantage when he hooked the ball wide from six yards.
The Cobblers continued to live dangerously and moments later Harrison squandered another good opportunity, firing wide from Deering's near-post cross.
Chris Hackett twice shot over for Cobblers but in between Matt Duke came to Northampton's rescue when he foiled Terry Gornell after he was released by Deering, before also denying Brown from close range.
Gornell squandered two more chances, with Duke twice denying him, as Cheltenham continued to pose a threat but in the end they had to settle for a share of the spoils.