Elliott's 44th-minute strike prolonged the Robins long unbeaten home record, which stretches back to September.
Adebayo Akinfenwa, leading scorer for the visitors, headed wide from a good position in the 22nd minute after a cross from Ishmel Demontagnac but Cheltenham edged the first half in windy conditions.
Skipper Elliott smashed the ball home from close range after seeing his initial header well saved by Lee Nicholls after Keith Lowe's cross from the right.
The Robins deserved their half-time advantage, but Northampton improved after the break and put the hosts under some pressure.
Scott Brown blocked a shot from Roy O'Donovan, who raced through on goal after a long clearance from Nicholls caught out the Cheltenham defence.
Wingers Chris Hackett and Demontagnac also went close to scoring an equalising goal, but a defence marshalled superbly by the outstanding Elliott stood firm to shut out Aidy Boothroyd's side.