The ban on Russia's track-and-field team going to Rio was imposed last November by the International Association of Athletics Federations (IAAF) after an independent report uncovered rampant state-sponsored doping in Russian athletics.
"It's a landmark statement that allows us to move forward and be inspired by pictures we'll see at the Games," Campbell, who was part of Britain's gold medal-winning 4x100m relay team at the Athens Games, told the British media.
"Any other decision would have been the death of athletics and the Olympics."
The decision by the CAS, sport's highest tribunal, increases the possibility of the International Olympic Committee (IOC) excluding Russia from all sports, not just track and field, in Rio de Janeiro.
(Reporting by Ian Rodricks in Bengaluru; editing by Sudipto Ganguly)
