Former Proteas skipper will miss much of the 2017 schedule due to long-standing elbow injury.
India were reduced to 63-4 but the captain led them back to take win.
Eoin Morgan and Jos Buttler both produce golden ducks as the tourists slump to heavy warm-up defeat.
35-year-old resigned as his country's ODI and T20 captain last week.
Former Ashes-winning spinner will head back to Edgbaston in the new year as sport director.
Kohli was the leading run-scorer in the recently concluded Test series against England.
Without world-class spinners, winning in sub-continent conditions is going to be hard, the England Test skipper said.
Cook will return home and hold talks with director of cricket Andrew Strauss over role.
Tourists experience another woeful batting collapse to lose the fifth Test by an innings and 75 runs.
India and England are currently in action in the fifth and final Test match at Chennai.
Hosts record highest score of 759 for 7 as England are embarrassed in fifth Test in Chennai.
Morgan has been named interim captain of Sydney Thunders in Shane Watson's absence.
However, India openers KL Rahul and Parthiv Patel mount an impressive response on day two of the fifth Test.
Moeen and Joe Root pile on 146 after two early wickets in Chennai.
Alastair Cook to discuss his position with Andrew Strauss next month.
The 34-year-old is suffering from 'body soreness' says captain Alastair Cook.
Inzamam believes that runs are runs irrespective of the conditions on offer.
The Adelaide Oval will host the first Ashes Test under floodlights in cricket history.
Kohli has scored 640 runs so far in the Test series at an average of 128.
England are struggling in the Test series against India and trail the hosts 2-0.
Hosts India lead the five-match Test series 2-0.
India close day two of the fourth Test trailing by 254 on 146-1 after a second-wicket stand of 107.