The 30-year-old is the third Indian to win the Sir Garfield Sobers trophy and is also ICC Test cricketer of the year.
Alastair Cook was named captain of the ICC Test team of 2016. The voting period closed in September, before England's Test series defeat in India.
Joe Root, Jonny Bairstow and Ben Stokes also feature but India's Virat Kohli and Australia's Steve Smith missed out.
Ashwin took 48 wickets and scored 336 runs in eight Tests, as well as 27 wickets in 19 T20 internationals.
"It feels great to follow Sachin Tendulkar and Rahul Dravid in being ICC Cricketer of the Year," said Ashwin.
Ashwin dedicated the award to his family and believes India "had a great transition" and "fell on the right track" since Kohli took over as captain following MS Dhoni's retirement in 2014.
ICC chief executive David Richardson said Ashwin "is justifiably now being counted among the best spinners of his generation".
Ashwin, who made his Test debut in 2011, now has 248 wickets from 44 Tests, and also averages 34.92 with the bat.
He joins Jacques Kallis (2005), Ricky Ponting (2006), Kumar Sangakkara (2012), Michael Clarke (2013), Mitchell Johnson (2014) and Smith (2015) by winning both the ICC world player and Test cricketer awards in the same year.
The ICC's voting period covers play from 14 September 2015 to 20 September 2016.
The ICC's other 2016 award winners are:
England skipper Cook was named as captain for the ICC's Test team of the year after guiding his side to an away win over South Africa and defeating Sri Lanka and drawing with Pakistan at home during the voting period.
The 31-year-old's role for England is under scrutiny after he said he would "go home and do some thinking" after the 4-0 Test series defeat by India that followed the historic Test loss in Bangladesh in the first part of the winter programme.
Smith is named only as 12th man while Kohli does not appear at all, with New Zealand's Kane Williamson, Root and Adam Voges of Australia filling the middle-order berths.
The Australia and India captains are currently ranked first and second respectively in the ICC's own Test batting ranking.
AB De Villiers - who has stepped down as South Africa captain while he recovers from injury - has been selected in the ODI team for the sixth time since 2010.
De Villiers and ODI player of the year De Kock are joined by compatriot Kagiso Rabada in the side while Jos Buttler is the only Englishman to feature.

Surrey Police said two teenage girls were waiting at a bus stop half way along Rydens Road, in Walton-on-Thames, when a man approached them and touched one of the girls on the bottom.
He then went into some nearby bushes and exposed himself.
He is described as white, in his mid 20s and about 6ft tall. He had short ginger or blonde hair and wonky teeth.
The incident happened at 19:30 BST on Thursday.
Det Con Ben Kirby said: "Prior to this incident, the girls had been travelling on the 461 bus to Walton town centre. However, it had broken down so they were waiting for another bus to continue their journey.
"We would like to speak to anyone who was on this bus as they might have information which will help us to identify this individual."
India spinner Ravichandran Ashwin has been named the International Cricket Council's world player of the year.
[NEXT_CONCEPT]
A man targeted two teenagers waiting at a bus stop by sexually touching one and exposing himself in some nearby bushes.