"We condemn this criminal activity and have asked the Russian Government to do everything in their power to make it stop," said Olivier Niggli, WADA's director general.
Niggli said the hack on WADA's computer system constituted "retaliation" against the agency, which imposed heavy penalties against Russia over doping, including banning practically its entire athletics team from the Rio Olympics.
But Russia denied involvement in the hacks and said the Russian government was ready to help if needed.
"Russia consistently backs fighting cybercrime, consistently invites all states and international organisations to cooperate in this area, and this position of Russia is well known," Peskov said.
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On Tuesday, WADA announced that the Russian cyber-espionage group Tsar Team (APT28), also known as Fancy Bears, had broken into its Anti-Doping Administration and Management System (ADAMS) database.
The hacking group released information gleaned from the files of sports stars including US Olympic gymnast Simone Biles and tennis champions Venus and Serena Williams.
"To those athletes that have been impacted, we regret that criminals have attempted to smear your reputations in this way," said Niggli.
Russia's Sports Minister Vitaly Mutko brushed aside suggestions of an orchestrated hacking campaign.
"How can you prove that they are hackers from Russia? You blame Russia for everything," Mutko said.
Foreign ministry spokeswoman Maria Zakharova said it was wrong to connect the issue of cybercrime with Russia's attempts to overcome the massive doping scandal around its athletes.
She accused WADA of sowing confusion, saying that "no one understands the principles behind the actions and functioning of this organisation".
WADA needs to "develop clear criteria for its work that everyone understands, based on international law, and not on some very strange statements," she said.
She reiterated that Russia denies any role in the hacks, saying: "For Russia just like for the rest of the world, hackers and hacking are outside the law."
"We can influence this process using laws, but we have no other way to influence the hackers," she stressed.
"I've openly discussed my TUEs with the media and have no issues with the leak which confirms my statements," the 31-year-old said.
"In nine years as a professional I've twice required a TUE for exacerbated asthma, the last time was in 2014."
TUEs exist to allow athletes with recognised medical conditions such as asthma, from which Rio gold medallist Wiggins suffers, to take drugs on WADA's banned list for their ailment so they can compete in elite level sport.