Kolkata failed to capitalise on a great start in reply to Chennai's 192/3 as the MD Dhoni-led side powered to a fourth IPL title in Dubai.
From a comfortable position of 96/1, KKR dragged themselves into a difficult situation in the end. They will play CSK in the final on Friday.
During last week's IPL match between Delhi Capitals and Kolkata Knight Riders, which is captained by Morgan, Ashwin tried to take a run after a throw ricocheted off his batting partner Rishabh Pant's body.
In a series of tweets, Ashwin clarified his stand and made it clear that he would again run if the ball hit a fielder's body.
Delhi skipper Pant and Ashwin had picked up a second run when the ball thrown by the fielder ricocheted off the batter and that seemed to have irked the KKR outfit.
Steve Smith and Rishabh Pant top-scored with 39.
The rookie opener smashed 53 off 33 balls as KKR outplayed defending champions Mumbai Indians by seven wickets.
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.