The BBC reported that the man who succeeded — his name was not given due to curbs on reporting around the case — had sought the deletion of search results from Google about a crime he had committed in the past.
The European Union's Court of Justice ruled in May 2014 that European law gave people the right to ask search engines to remove results for queries that include their name.
UK High Court Justice Mark Warby ruled in his favour, but turned down a claim by another businessman who had committed a more serious offence.
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The second man was jailed for four years more than a decade ago after being convicted of conspiring to account falsely.
Both had approached Google and asked the company to delete the search results about their convictions, including links that led to news stories, arguing that this material was no longer relevant.
The pair took the issue to court when Google refused to pay heed to their requests.
Google said in a statement that it would respect the rulings.
"We work hard to comply with the right to be forgotten, but we take great care not to remove search results that are in the public interest," the company said.
"We are pleased that the Court recognised our efforts in this area, and we will respect the judgments they have made in this case."