While Google, Bing and other search engines can remove links from their search results, they cannot delete the news articles or web pages.
Google had refused to remove the links, leading NT2 and another man known as NT1 to take the company to court.
A businessman fighting for the "right to be forgotten" has won a UK High Court action against Google.
Mr Justice Warby said the businessmen, who can not be named for legal reasons, complained of results returned by Google Search that feature links to third-party reports about their convictions.
Mr Justice Warby delivered a blow to Google by ruling in his favour on Friday.
NT2, who won his case, had been sentenced to six months imprisonment for conspiracy to carry out surveillance.
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Although the judge ruled out any damages payments for the first businessman, the decision is a landmark case and will have implications on future requests from convicted criminals or embarrassed individuals who want their records scrubbed from Google.
Both men were challenging Google's refusal to remove some links which contained information about their criminal past. That man, referred to in the media as NT1, was convicted of conspiracy to account falsely in the late 1990s and spent four years behind bars.
NT2 had "frankly acknowledged his guilt", he said, adding: "His current business activities are in a field quite different from that in which he was operating at the time". In those circumstances, "the public interest in having information with his name about this case doesn't prevail".
"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 and will defend the public's right to access lawful information", the spokesperson said. "The right to be forgotten is meant to apply to information that is no longer relevant but disproportionately impacts a person", said Jim Killock, executive director of The Open Rights Group, which fights for internet freedoms. We are pleased that the court recognised our efforts in this area, and we will respect the judgments they have made in this case.
Google reported in February of 2018 that from the period between 2014 to 2017, the company received over 2.4 million requests to delist URLs under the right to be forgotten statute. The two linked cases are: NT1 v. Google and NT2 v. Google, High Court of Justice, Queen's Bench Division, Case No.'s HQ15X04128 and HQ15X04127. Others, however, expressed concern that the ruling could harm free press and speech rights.