Kohli's men topped the group stage and wrested the number one ranking from England before their last four clash with 2015 finalists New Zealand.
Defending 239-8, Matt Henry and Trent Boult blew away India's star-studded top order that included Kohli and Rohit Sharma, the tournament's leading scorer, to reduce them to five for three wickets inside four overs.
Ravindra Jadeja mounted an audacious rearguard action but India could not recover after slumping to 24-4 after the first powerplay.
"Of course, very disappointed," a glum Kohli told a news conference.
"We've played outstanding cricket throughout this tournament. To just go out on the basis of 45 minutes of bad cricket is saddening.
"It breaks your heart also, because you worked so hard throughout the tournament to build momentum. You finish number one in the table, and then a spell of bad cricket and you're out of the tournament. But you have to accept it."
After the top order imploded, Rishabh Pant and Hardik Pandya tried to arrest the free-fall before both played reckless shots to throw away their wickets.
Kohli conceded their shot selection could have been better though he refused to blame 21-year-old Pant, who was playing only his ninth one-day international.
"Look, he's an instinctive players. And he did well to overcome that situation and string in a partnership with Hardik," Kohli said.
"He's still young. I made many, many errors when I was young in my career, and he will learn. He will look back and think yes, he could have chosen a different option in that situation, maybe and he realises that already."
India suffered their only group stage defeat against hosts England, who meet Australia in the second semi-final in Birmingham on Thursday.
Kohli said the team would emerge stronger from the setback.
"It's happened to us before and we have all come out better cricketers because of the setbacks, especially at a stage like World Cup," he added.
"But having said that, we have qualified to the semis and have played some really good cricket. So we should be also proud of the way we played."
