It beats the previous record of 73.6 million, which was set during the London 2012 Olympics.
The 2014 FIFA World Cup Brazil, Wimbledon and the British Grand Prix were among the most popular events with browsers.
BBC Sport also experienced its highest weekly mobile reach, welcoming 8.1 million unique mobile browsers in the first full week of the World Cup.
In the period beginning with the World Cup on June 12 and ending with last week's Open Championship, BBC Sport attracted 96 million unique browsers at home and abroad.
Approximately 37.8 million of the monthly users were based outside of the UK, helping the site extend its international reach.
Video viewership was also up, with the World Cup alone resulting in 15.9 million live video requests.
Elsewhere, the BBC Sport Facebook welcomed 1.25 million new fans during the World Cup, while its Twitter page is closing in on 2.5 million fans.
BBC Sport's Chris Condron said that the company has been evolving its technology since the Olympics, adding even more interactivity.
"There's plenty more to come from BBC Sport this summer with the Commonwealth Games and the new football season, which features the return of the FA Cup to the BBC," Condron continued.
"We'll be ensuring audiences can get all the action they want, wherever and whenever they want it."
The 2014 Commonwealth Games opening ceremony was seen by 7.6m. The World Cup final was watched by a peak audience of 16 million.