Kohli, who recently retired from Test cricket, was in tears as Bengaluru finally shed the underachiever's tag that has stuck to them since the inaugural 2008 edition of the competition.
Bengaluru lost IPL finals in 2009, 2011 and 2016, but finally lifted the trophy following a thrilling six-run win over the Ricky Ponting-coached Punjab Kings in Ahmedabad on Tuesday night.
"It probably means everything to (Kohli)," Hazlewood said.
"Some of us come and go in the franchises but to be there from the start and to slug it out for 18 seasons and to get a result, I think we'll see a bit of emotion there."
Kohli, who top-scored with 43, told broadcasters what it meant to him: "It's been 18 long years. I've given this team my youth, my prime and my experience, and I've tried to win this every season.
"To finally have this moment come true, it's an unbelievable feeling. I never thought this day would come. I was overcome with emotions as soon as the last ball was bowled."
Kohli has been with Bengaluru since 2008, including nine seasons as captain. The relief was palpable as he hugged former teammate AB de Villiers.
"I've stayed loyal to this team no matter what," he said.
"I always dreamt of winning it with them, and this is far more special than winning it with anyone else.
"This is the team I'm going to play for till the last day in IPL."
Hazlewood, who won an IPL title with Chennai in 2021, claimed 1-54 in the final to finish with 22 for the tournament.
The 34-year-old was the third-highest wicket-taker of this IPL, despite missing two games due to injury, only behind Indian quick Prasidh Krishna and Afghan young gun Noor Ahmad.
Fellow Australians Josh Inglis (39) and Marcus Stoinis (six) both ended up on the losing side with Ponting for the Kings.
Hazlewood and Inglis will immediately fly to England to prepare for the WTC final against South Africa at Lord's, which starts on June 11.
A key member of Australia's bowling attack, Hazlewood hasn't played a Test since December when he broke down with injury against India at the Gabba.
Scott Boland performed superbly in his absence, but the recent form of the 72-Test veteran could be impossible for selectors to ignore reuniting him with Pat Cummins and Mitchell Starc.
Inglis played in both of Australia's Tests against Sri Lanka in Galle earlier this year, smashing a century on debut.
But the Western Australian is no certainty to be in the XI next week as fit-again Cameron Green looks likely to return as a batter-only after marking a successful comeback from back surgery.
With Reuters.
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