Country's white-ball skipper Eoin Morgan and wicketkeeper-batsman Jos Buttler are under investigation by the ECB after their tweets, mocking Indians, surfaced on the social media on Wednesday.
Leg-spinner Rahul Chahar picked up 4/27, while Jasprit Bumrah and Trent Boult were too good for the KKR batsmen at the death.
Batsman Liam Livingstone is set to make his ODI debut in the second match on Friday.
Jofra Archer (3/23) and Mark Wood (1/20) conceded just 43 runs in eight overs picking four wickets between them as India were restricted to just 124/7 in their 20 overs.
England overhauled the modest target of 125 in just 15.3 overs to take a 1-0 lead in the five-match series.
Malan and Morgan compiled a 51-run partnership to haul England back into the game after a career-best 3/19 spell for home spinner Tabraiz Shamsi had South Africa well placed to level the series.
The result also meant defending champions Mumbai Indians qualified for the play-offs.
Varun Chakravarthy claimed 5/20 as KKR won by 59 runs.
After the bowlers restricted KKR to 148/5, opener Quinton de Kock blazed his way to an unbeaten 78 off 44 balls to complete the task for Mumbai.
Chasing a modest target of 149, Mumbai were home and dry with 19 balls to spare.