CAS denied an appeal by Sweden asking for a gold to be awarded to silver-medallist Norden as well as Switzerland's Nicola Spirig, who was declared the winner in a photo-finish last Saturday.
Spirig and Norden clocked identical times of one hour 59 minutes 48 seconds in one of the closest triathlon finishes seen.
The Swedish Olympic Committee and Swedish Triathlon Federation (STF) asked for both athletes to be ranked as joint winners instead of just Spirig.
But after a 3-1/2-hour meeting by a three-person panel on Friday, CAS denied the application because it said there was no violation of rules and the matter concerned a "field-of-play decision".
According to long standing CAS jurisprudence, a "field-of-play decision" is unable to be reviewed by CAS panels except in the event that there has been arbitrariness or bad faith in arriving at the decision.
Sweden had appealed after the race but the International Triathlon Union (ITU) rejected it on Wednesday.
Judges examined the photographic evidence after the race and decided Spirig's winning margin was less than 15 centimetres. Officials later issued a photograph which appeared to show Norden's head crossing the finish line first.
However, the ITU ruled that the winner was the athlete whose torso crossed the line first and said two cameras at the finish showed the Swiss athlete's torso ahead of the Swede's.
It was the first time that an Olympic triathlon had been decided on a photo-finish.