Mangs' lawyers had based their application to the Supreme Court on weapons analyses, which tied Mangs' weapon and bullets to the crime scenes.
The so-called Malmö sniper wanted the analysis to be repeated by technicians not previously engaged in the case.
The ruling in Sweden's highest court puts a final end to what Malmö residents called a period of terror as an unknown gunman targeted mostly foreign-born victims in the southern city with a Glock 19 pistol.
Mangs was arrested in November 2010 after a string of shootings that took place in Malmö against people of immigrant origin.
He was charged with killing two men of immigrant origin, aged 23 and 66, in 2003 and a 20-year-old Swedish woman who had been sitting in a car with an immigrant man in 2009.
He was also charged with a slew of attempted murders in which he fired numerous shots with his Glock 19 pistol at homes, businesses and cars as well as out in the open, seriously injuring a number of people and coming close to killing many others.