The Court of Arbitration for Sport in Lausanne said the evidence presented against the trio showed that the charges were "established beyond reasonable doubt and that the sanctions imposed should be upheld."
In January of last year, the ethics commission of governing body the IAAF barred the group from any involvement in athletics, alleging that they took bribes to cover up doping by Russian athletes.
Balakhnichev had been the head of the All-Russia Athletic Federation (ARAF), who supervised Melnikov, a former chief ARAF coach for long-distance walkers and runners.
The former ARAF boss was furious with the decision, although admitted he hadn't expected the appeal to be successful.
"I didn't expect any other decision. Everything that is currently taking place in the sports legal system is complete rubbish," Balaknichev told Russian press agency R-Sport.
"It's an unfair, absolutely unlawful decision."
Papa Massata Diack, a former marketing consultant to the IAAF, is one of 15 sons of the fallen former president.
The IAAF sanction was imposed just before the World Anti-Doping Agency (WADA) released a bombshell report detailing a massive doping programme in Russian track and field.
WADA claimed the cheating had state support in Moscow and that IAAF leaders must have known about it, given the wide scope of the doping.
The scandal led to dozens of Russian athletes being barred from the Rio 2016 Games.
Moscow has fiercely denied the existence of a state-sponsored doping programme, but has taken steps to get Russian athletes back into international action, including by setting up an independent anti-doping lab.
Separately, Papa Massata and Lamine Diack also face charges in France over millions of dollars paid to cover up doping failures by Russian athletes.