The increase, from 26.2 million to 29.4 million, is the biggest percentage change in month-on-month traffic for all the UK newspaper websites included in the ABC report.
The Guardian was the only other website to record positive growth with a month-on-month increase of 2.4 per cent, to 84.9 million, making June the fourth month in a row in which Guardian.co.uk has broken its own record for web traffic.
Much of the traffic to the Guardian's website was fuelled by "explosive NSA revelations", said David Pemsel, chief commercial officer at Guardian News and Media, in a press release. The article naming Edward Snowden as the source received 3.7 million page views on the day it was published, making it the most viewed article in Guardian.co.uk's history, and the NSA content on the whole was viewed by 8.8 million unique browsers.
The London Evening Standard and Independent, which recorded the two highest increases in month-on-month traffic for May, both saw a slump in unique visitors in June with figures falling by 12.3 per cent and 11.5 per cent respectively.
Traffic to the the Mail Online fell by 6 per cent to 120 million, while traffic to the Telegraph fell by 5.4 per cent to 54 million.
The Metro's online traffic fell by 1.8 per cent to 9.6 million, although remained far higher than the 5.2 million recorded at the end of last year as readers adjusted to the site's responsive redesign. Online traffic to the Sun also fell month-on-month by 1 per cent.
Update: A spokesman for Trinity Mirror said: "Since we integrated our print and digital teams earlier this year our growth has been hugely encouraging. We are investing in the digital team and our journalism which is attracting new users all the time. Additionally, by working hand-in-glove with editorial the product team have delivered innovative new formats and functionality across our sites.
"In the last three months we have led the news agenda with a series of exclusives including our inside North Korea footage, the exclusive of the police apprehending the Woolwich terror suspects, and, of course, the recent Nigella scoop, which was a global news phenomenon."
This article has been updated to clarify that this was the fourth, not fifth, record month in a row for the Guardian in terms of web traffic. In terms of mobile traffic, June was the fifth record month in a row for the Guardian.co.uk.