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Ali took 10-112 in his first 10-wicket haul and Cook made an important contribution of 69 to help England win
England will take on an AB de Villiers-less South Africa in the European side's first ever test encounter after former skipper Alastair Cook stepping down.
Joe Root compiled a composed, unbeaten 90 while all-rounder Chris Woakes hit an undefeated 68 as England completed a four-wicket victory.
The new England skipper first came across Stokes in an Under-12s tournament when the all-rounder was representing Cumbria.
Concerns have been raised about whether the burden of captaincy will reduce Root's effectiveness with the bat.
The Yorkshire batsman had been the overwhelming favourite to replace Alastair Cook ever since the Essex opener resigned as Test skipper.
Root is currently England's vice-captain and is the overwhelming favourite to replace Alastair Cook, who resigned as Test skipper on Monday.
Cook, England's leading scorer in Tests with 11,057 runs, became captain in August 2012 and led his country to Ashes glory on home soil in 2013 and 2015.
Umpire C Shamsuddin had adjudged Root leg before wicket off Jasprit Bumrah in the last over when the batsman had actually edged the ball.
The hosts made a solitary change to the team with Ajinkya Rahane replacing opener Shikhar Dhawan, whose form has been patchy.
England, led by limited-overs captain Eoin Morgan, are scheduled to play two further ODIs and three Twenty20s internationals in India.
Kohli and Jadhav rattled up a sensational 200-run partnership to set up the platform after England set the hosts a stiff 351-run target.
Root, who had stayed back for a few days for the birth of his first child, is not playing the second warm-up game against an India A.
England batsman Joe Root urged today's centurion Moeen Ali to put India under pressure.
Joe Root's classy 88 and Moeen Ali's unbeaten century (120 batting) helped England recover from 21 for 2 and post a modest 284 for 4.