Inglis scored an important 38 runs for Punjab in Sunday night's second qualifier, helping the Kings to a five-wicket win over the Mumbai Indians in Ahmedabad.
The result sent the Ricky Ponting-coached franchise into their first final in 11 years, where they will face Hazlewood's Royal Challengers Bengaluru after they qualified last week.
Marcus Stoinis will also feature in the decider for Punjab, giving Australia three players and a high-profile coach in the biggest T20 franchise game of the year.
But it is Hazlewood and Inglis' qualification that will have the biggest impact on Australia's Test squad.
The IPL final was initially scheduled for May 25, before the week-and-a-half postponement due to India's conflict with Pakistan prompted it to be pushed back.
Australian players not in the IPL flew out to the United Kingdom last week ahead of the championship bout against South Africa, which will begin at Lord's on June 11.
It's now expected Hazlewood and Inglis will fly out of India the day after Tuesday night's final, giving the pair six days to prepare once they touch down in London.
Inglis is facing an uphill fight to be part of Australia's Test XI despite scoring a century on debut in Sri Lanka, with Cameron Green's return creating a squeeze.
Hazlewood is also in a selection battle.
The 34-year-old last played a Test in December, before a calf strain ruled him out of the finish to the home India series and Sri Lanka tour.
In his absence Scott Boland again impressed, with a career-best haul of 10-76 to help Australia reclaim the Border-Gavaskar Trophy at the SCG.
Selectors face a tough decision on who to pick out of the pair, with Ponting this week predicting Hazlewood will regain his spot for Lord's.
The veteran quick had battled a shoulder niggle that at one stage threatened to rule him out of returning to the IPL, before eventually being cleared.
The Proteas will not carry the same challenges into next week, after South Africa Cricket officials ordered all players out of the IPL by the initial final date of May 25.
Quicks Marco Jansen and Lungi Ngidi would otherwise have been due to feature in the decider.
It comes as South Africa also called in Australia's Ashes arch-nemesis Stuart Broad to act as a consultant ahead of the World Test Championship final.
Meanwhile, Hazlewood's Bengaluru side will also be hunting their first IPL crown after losing the final in 2009, 2011 and 2016.
To do so, the Virat Kohli-led Royal Challengers will need to beat a Punjab side they disposed of in the first match of the playoffs last week.
They will also need to find a way to stop in-form Kings captain Shreyas Iyer, whose 87 from 41 balls helped his side chase Mumbai's 6-203 with an over to spare.
Inglis' 38 was crucial in the context of the match, with the West Australian blasting five fours and two sixes in his 21-ball knock to give his side a platform in the powerplay.
