The Terriers added to the hosts' problems as their terrible recent home record was extended to 11 games without a win.
Norwood had made the breakthrough in the 10th minute with a well-taken finish following an effervescent start by Mark Robins' side, before Clayton smashed home on the hour as Birmingham failed to provide any sort of test despite Olly Lee's very late consolation.
While the home side came into the game with just one win in seven league games, the Terriers were in similarly poor form after losing five of their previous six in all competitions.
Birmingham started well but had a fright in the fourth minute when goalkeeper Darren Randolph failed to punch away a corner and a lucky bounce off Huddersfield's Anthony Gerrard went just over.
Town's Danny Ward, in for the injured James Vaughan, sent a left-footed shot from 30 yards fizzing past Randolph's left-hand post in the seventh minute.
Huddersfield were ahead in the 10th minute and Ward was the architect, chipping a delicate left-footed pass to the feet of Norwood, who controlled well before bending home a cool finish.
Town's Clayton was booked for a crude late challenge on Emyr Huws as Birmingham struggled to get going again.
They had a chance in the 26th minute when midfielder Tom Adeyemi did well to meet Chris Burke's cross from deep, only to see his header go just wide.
Ward hit a tame shot just after the half-hour mark before Burke, at the other end, failed to get enough on his left-footed shot to really test Alex Smithies.
Smithies was then booked after being warned on several occasions for time-wasting by referee Darren Deadman before the half was even out.
Huws forced a rare chance for Birmingham 10 minutes into the second half but his shot was straight at Smithies. Nikola Zigic was given some rare playing time by Birmingham boss and former Huddersfield manager Lee Clark, replacing Adeyemi in the 59th minute and arriving to a mixed reception from the Blues fans.
A minute later it was 2-0, the impressive Clayton capitalising on far too much time and space to drive forward and smash home a powerful drive from 25 yards which Randolph failed to get anywhere near.
Moments later City somehow failed to score when Burke's cross was not dealt with and Federico Macheda only managed to swing at the ball and miss.
The home fans' dismay at five minutes of added time summed up their afternoon, though substitute Lee gave them something to cheer after smashing home on the volley following Zigic's header in the box.