The young striker fired home a loose ball from a corner on the half-hour mark as the Canaries recovered from their unlucky defeat at Leeds on Monday with a hard-fought home success.
Swindon competed well but never looked like taking anything from the game against a Norwich side who made it five wins in six matches.
After a slow start, Norwich finally got the goal their growing pressure merited when poor defending from a corner allowed Martin to fire home from close range. Top scorer Grant Holt had earlier hit the crossbar with a shot from the edge of the area.
Swindon were on the back foot for long periods, though they might have equalised on the stroke of half-time when Anthony McNamee slid the ball wide after Temitope Obadeyi got the better of George Francomb.
Defender Jens Berthel Askou headed just wide after the break from Darel Russell who took on Kevin Amankwaah while at the other end a shot-shy Swindon side failed to trouble keeper Fraser Forster.