Two goals in the space of seven first-half minutes from David Silva and Gabriel Jesus appeared to have put City in control of the contest, but Leicester responded through Shinji Okazaki to give themselves hope heading into the interval.
The Foxes had the chance to complete the comeback in the second half when they were awarded a penalty, but Riyad Mahrez inadvertently hit his spot kick twice and the goal was subsequently disallowed.
City's win sees them climb up to third in the Premier League table, now six points clear of fifth-placed Arsenal ahead of the Gunners' trip to Stoke City later this afternoon.
Leicester - dethroned as Premier League champions courtesy of Chelsea's win over West Bromwich Albion on Friday night - were two goals ahead inside the opening five minutes of the reverse fixture at the King Power Stadium in December, but they were almost on the other end of a quick start this afternoon when Fernandinho nodded a fourth-minute header wide of the target having been left unmarked inside the area.
The Foxes created their own chance from a corner shortly afterwards as the match got off to a quick start, but Wilfred Ndidi was also unable to make the most of space inside the area as he put his header well off target.
City soon began to take control of the contest, though, and Leroy Sane squandered another clear opening for the hosts when he skewed a left-footed strike across the face of goal when he would have been better off going with his right.
Sane had another sight of goal later in the half, but only after Gael Clichy had turned down the opportunity to shoot following an important clearance from Mahrez in his own six-yard box.
City did take the lead shortly before the hour mark, though, with Sane the provider on this occasion as Silva steered the German winger's low cross into the bottom corner with his first-time finish.
It was the first time that Silva had scored in back-to-back league games since December 2014, but Leicester felt that the goal should have been disallowed after Raheem Sterling tried to get the final touch from an offside position.
Referee Bobby Madley allowed it to stand, though, and the official gave another damaging decision against the Foxes seven minutes later when he pointed to the spot after Yohan Benalouane brought down Sane inside the area.
The referee had little option but to award the penalty following the Leicester defender's reckless challenge, and Jesus thrust himself ahead of birthday boy Yaya Toure in the queue to tuck a composed spot kick into the bottom corner.
City appeared to be in complete control at that stage of the match, but Leicester gave themselves a lifeline shortly before half time when Okazaki ended a 23-match goal drought in style with a stunning first-time volley to fire Albrighton's cross past Caballero.
That goal seemed to give Leicester renewed belief for the start of the second half, and they began brightly with Albrighton failing to get a clean connection on Jamie Vardy's cross after losing his marker inside the area.
City, by contrast, began the second half slowly, but they eventually began to settle into the game again and missed a good chance to restore their two-goal lead with 20 minutes remaining when Jesus failed to pick out an unmarked Sterling in the box moments before Toure curled an effort into the arms of Kasper Schmeichel.
Sane then brought another stop from the Leicester keeper before Sterling was released down the opposite channel, only for Benalouane to get back and make a crucial block.
The Foxes were given a great chance to restore parity and dent City's top-four hopes with less than 15 minutes remaining when Clichy clumsily brought Mahrez down just inside the area, leading to the second penalty of the game.
Mahrez slipped while taking the spot kick, though, and while the ball ended up in the back of the net the goal was disallowed as the winger inadvertently touched it twice before it flew past Caballero.
Vincent Kompany and Vardy failed to make the most of half-chances in the final 10 minutes as both sides searched for the all-important fourth goal, and the City nerves were tested even more when the fourth official indicated a minimum of seven minutes added time.
In the end it went on for a further 10 minutes, but despite Schmeichel going up for a late free kick there were to be no further goals as Man City held on for a vital victory in their bid for a top-four finish.
City's superior goal difference means that defeat for Arsenal against Stoke later today would all but guarantee their place in next season's Champions League, otherwise City would effectively only need a win from one of their final two fixtures against West Bromwich Albion and Watford to clinch a top-four spot.
Leicester, meanwhile, remain unchanged in ninth place having seen a two-match winless streak come to an end as they suffered just their second defeat away to Man City since January 1977.