The increase, of 7.9 per cent month-on-month to 29.9 million, represents the highest ever monthly audience for the title.
"We've had an overwhelmingly positive year," Christian Broughton, digital editor at The Independent, told Journalism.co.uk, detailing how improvements in breaking news coverage and the "shareability" of content and social media had played a large role in the increase.
"I think our content is playful, entertaining, informative and high quality," he said, "and it maintains the core values of the Independent".
The Evening Standard saw a significant increase in traffic as well, with monthly visitors rising 6.5 per cent to 4.5 million.
Other titles to see an increase in monthly visitors were Mirror Group Digital, up 5.3 per cent to 42.9 million; and The Guardian website, up 1.6 per cent to 85.3 million.
The Telegraph also experienced a small increase of 0.6 per cent to just under 61 million.
The Daily Mail and Metro were the only titles to see a decrease in the number of monthly visitors. Traffic to the MailOnline fell by 4 per cent to 161.3 million, still the highest by some distance, and at the Metro the fall was by 2.8 per cent, to 19.5 million.
Year-on-year however, the Metro had the largest increase in traffic of 274 per cent, from 5.2 million in December 2012 to 19.5 million in December 2013.
"2013 was a big year for Metro.co.uk," Dominic Rowell, chief customer officer at Metro, told Journalism.co.uk, "starting with our site redesign and ending with this growth.
"These results are testament to a team and a technology platform that's really clear about what our audience needs – short doses of stimulating content throughout the day. We're confident of carrying this momentum into 2014."
"Social growth" also played a large part in the increase according to Richard Moynihan, social media and community manager at Metro, who said social referral traffic "has grown 17.5 times" bigger over 2013 and 19 times more referrals from Facebook.
Mirror Group Digital, which launched UsVsTh3m and Ampp3d in 2013, also had a good year with an increase of just under 132 per cent, from 18.5 million to 42.9 million, although UsVsTh3m is hosted on a separate Tumblr blog, rather than under a Mirror domain.
The Independent saw annual increases of just over 40 per cent, a figure Broughton called a "pretty bold achievement" when compared to the growth of other, larger news outlets.
The Independent has "huge ambitions for this year ," he said. Although Broughton declined to go into details at this early stage he said new plans will be announced throughout the year "which we hope will give us new growth".
Traffic to Mail Online increased by 41.5 per cent, while the Indpendent's sister site Standard.co.uk also increased overall in 2013, by 37.7 per cent.
The Guardian's web traffic increased by 24.3 per cent year-on-year, while the Telegraph website received the smallest increase in traffic over 2013, at only 7.7 per cent.
Last week the Guardian reported that Alexander and Evgeny Lebedev are looking to sell the Independent titles, including the Independent on Sunday and i.
Correction: This article has been corrected to show that overall social referral traffic to the Metro website has grown 17.5 times from December 2012 to December 2013, with 19 times growth in Facebook referrals, not 19 times growth in overall social referrals, as previously stated.