After a fraught opening period on Tuesday night, Okazaki latched onto Vicente Iborra’s header after 65 minutes to spark new life into City and put them ahead before Slimani’s curler late on.
Craig Shakespeare made seven changes to the line-up which drew 1-1 at Huddersfield Town in the Premier League at the weekend, handing a debut to Austria defender Aleksandar Dragović.
In front of a full house on Filbert Way, the Foxes were up against an experienced Reds outfit who - with Philippe Coutinho back among their ranks - looked dangerous at every turn in attack.
Inside 10 minutes, captain Wes Morgan and Academy product Ben Chilwell had to be alert to the Brazilian’s eclectic talents with a succession of last-ditch challenges to deny him at the back.
Goalkeeper Ben Hamer, deputising for Kasper Schmeichel, also impressed for the Foxes and it was his smart save which frustrated Coutinho once again early on at King Power Stadium.
While Jürgen Klopp’s eight-time holders had City worried on occasions, Demarai Gray and Marc Albrighton also threatened, cutting inside on the flanks and seizing upon loose balls.
Following the interval, Leonardo Ulloa sustained an injury while covering for his colleagues at the back, paving the way for Okazaki to continue his rich goalscoring form this season.
The Japan international, who recently helped his country qualify for the 2018 World Cup, made an instant impact for Shakespeare’s men, driving them forwards whenever possible.
His industrious nature paid dividends on 65 minutes when Iborra headed it down into his path before he twisted and turned in the box and picked his spot in Danny Ward’s bottom corner.
Buoyed by their one-goal advantage, City pressed forwards in numbers and they were rewarded for their attacking intent when Slimani located the top corner from 25 yards to make it 2-0.
Late on, in a move which contributed to the feel-good factor on Filbert Way, Shakespeare then handed a cameo to youngster Hamza Choudhury, who featured for eight minutes.
Key moment
It was a moment which flipped the match on its head and changed the direction of travel for City.
For much of the first half, it seemed a matter of time until Liverpool’s pressure would tell, but Okazaki’s opener just after the hour-mark put Leicester on the front foot.
Who impressed?
While Okazaki’s impact was key for the Foxes, Slimani’s finish to make it 2-0 was a moment which will live long in the memory for the Blue Army.
The skill required to find space and look up before executing a perfect finish from distance was a timely reminder of the Algerian's attributes in attacking positions.
What does it mean?
City will be in the pot for the Round 4 draw of the Carabao Cup, which you can follow on the Club’s official channels on Wednesday evening.
So far in the draw; Middlesbrough, Bournemouth, Norwich City, Bristol City, Crystal Palace, West Ham United, Leeds United, Wolves, Swansea City and Tottenham Hotspur.
Coming up…
For the second time in just five days, Klopp’s side return to King Power Stadium on Saturday for what is likely to be another tough clash in the Premier League (5:30pm kick-off).