Furnish - who with John is one of seven claimants suing Associated Newspapers, publisher of the Daily Mail, for alleged privacy breaches - also said in his witness statement on Thursday the Mail titles had been "actively homophobic" about him and John.
Associated denies allegations of phone hacking and other unlawful acts made by Furnish, John and the other claimants, and says the information its newspapers reported was already in the public domain or obtained from legitimate sources.
The publisher's lawyer, Catrin Evans, suggested information in the stories Furnish and John complained about had come from other media, John's website or statements from his spokesperson.
But Furnish, giving evidence by video link, said details in the Mail articles had been much more specific.
Furnish was the sixth claimant to give evidence in the high-profile lawsuit, after Harry, actors Elizabeth Hurley and Sadie Frost, former lawmaker Simon Hughes and anti-racism campaigner Doreen Lawrence. John is due to give evidence on Friday.
Furnish and John's case relates to 10 articles published between 2002 and 2015, including a 2007 story in the Mail on Sunday about a concert to mark the 10th anniversary of the death of Harry's mother Princess Diana, who died in a 1997 car crash.
Harry said in his witness statement he thought he and John would have exchanged voicemail messages about the concert. The Mail's lawyers, however, say the information came from well-placed royal sources.
Furnish said in his written evidence he and John believed their home's landlines had been bugged and the Mail was involved in "the stealing of our information through our friends Elizabeth Hurley and Prince Harry".
"Elton absolutely adores Prince Harry," his statement said.
Furnish also said he and John "have a long and difficult history with The Mail", accusing the publisher of writing "judgmental and narrow-minded stories about us".
Furnish said that, in relation to a 2010 article about him and John having a child through surrogacy, the Mail obtained a copy of their son's birth certificate before they did, which he said was "deeply disturbing".
"It has outraged us that The Mail used our friendships against us by stealing information through them," he added in his statement.
"It has outraged us that our family home was violated and our son's birth certificate stolen before we even had a chance to see it ourselves and Elton's medicals were ransacked with impunity."
