Chris Hughton's men were looking for back-to-back Premier League victories in order to alleviate the threat of relegation at Carrow Road.
However, they crumbled at the Liberty Stadium with a first-half double from Jonathan de Guzman and Wayne Routledge's strike with 15 minutes remaining doing the damage.
Almost 900 Norwich fans made the more than 600-mile round journey to and from south Wales, the longest trip Norwich supporters will make in the league this term.
Norwich full-back Russell Martin explained the squad would now repay the price of admission as a way of apologising for the below-par performance.
Speaking to the club's official website he said: "Our fans have been brilliant home and away this season and the numbers we get on the road are really impressive when you consider how far they have to travel up and down the country from Norfolk.
"All the lads really felt for the fans at the end of Saturday's game, with most of them facing something like a six-hour drive home. Hopefully this will help them with the cost of travelling to our next away game at Fulham.
Norwich are just six points above the relegation zone and, with crucial encounters at home to West Brom and away to Fulham in the next two league encounters, Martin has called on supporters to continue backing the side.
"These next few games are obviously vital and the support and the atmosphere at our recent win over Sunderland at Carrow Road made a massive difference," he added.
"So hopefully they'll get right behind us again this Saturday when we play West Brom, at Fulham and in all of our remaining games, to help us finish as high up the table as possible."
