Kelechi Iheanacho and Timothy Castagne scored in either half to cancel out Matheus Cunha's early opener and hand the Foxes their first win since February 11.
Leicester came into the game looking to snap their run of 10 consecutive games without a win across all competitions and move out of the relegation zone.
Newly-appointed manager Dean Smith received a baptism of fire in his first game in charge as the Foxes were beaten 3-1 by title-chasing Manchester City last Saturday.
Leicester came into the weekend on a four-game losing streak in the Premier League matches while claiming just one point from their last nine league outings since February's 4-1 victory over Tottenham.
This underwhelming run of results had seen them drop to 19th place in the league standings, two points away from safety and headed for the Championship for the first time since 2014.
As for Wolverhampton Wanderers, they steered out of danger at the bottom end of the table thanks to back-to-back home wins over London outfits Chelsea and Brentford in their previous two outings.
They journeyed to the King Power Stadium on a three-game unbeaten run — their longest run of games without defeat since December — and this upturn in form has seen them move to 13th place in the table, seven points above the relegation zone.
While Leicester set out to complete a top-flight double over Wolves for the first time since 1981, Julen Lopetegui's men came into the weekend looking to pick up a third league win on the bounce for the first time since January 2022.
It was nearly a dream start for the Foxes as Jamie Vardy, who was put through on goal, squared the ball across the box for Tete, but the Brazilian's low shot was blocked by Toti Gomes near the six-yard box.
However, after an early spell of possession, Leicester City had themselves to blame as they gifted the visitors the opener in the 13th minute thanks to another piece of poor defending.
Mario Lemina won the ball deep inside the Foxes' half after catching Youri Tielemans in possession before Atletico Madrid loanee Cunha drilled his low strike into the bottom-left corner.
Wolves continued to mount the pressure as Cunha's effort was blocked by Wout Faes at the near post before Dawson saw his venomous strike deflect narrowly behind for a corner.
Lopetegui's side, who are unbeaten in each of their last seven matches when scoring first, came forward again, but this time Pablo Sarabia's through ball to Diego Costa was an awful one as another promising attack went begging.
Leicester City found a way back into the game in the 37th minute courtesy of Iheanacho's cool penalty after Vardy was hacked down by the onrushing Jose Sa, who slipped while trying to close down the Englishman.
Buoyed by their leveller, the Foxes nearly took the lead on the stroke of half time as Iheanacho's curler from the edge of the box was initially fumbled by Sa before the Wolves keeper secured the ball at the second time of asking to deny Patson Daka an easy tap-in.
Leicester City picked up from where they dropped off in the first half as Kiernan Dewsbury-Hall, who came on for the injured Vardy, sent his shot from the edge of the box inches wide of the target.
Smith's men fluffed another glorious chance to take the lead in the 56th minute, this time Daka was put through on goal but fired his shot straight at Sa before Dewsbury-Hall fired his rebound over the bar from close range.
As Wolves regained control of the game, Leicester soaked up the pressure and hit back against the run of play, with Belgian international Castagne finding the back of the net with 15 minutes to play.
Both full-backs were involved in the goal as Victor Kristiansen galloped down the left before sending the ball into the Wolves box for Castage to guide past Sa and hand the hosts a huge goal in their survival battle.
It was late drama at the King Power Stadium as Daniel Iversen did well to tip Ruben Neves's late free-kick for a corner before the referee signalled play on after the VAR stopped the game for a potential Wolves penalty, with Faes judged to have touched the ball in the box.
Leicester City did well to see out the win and have now ended their run of eight consecutive Premier League games without a win when conceding the first goal.
The result has also seen the Foxes surge from the danger zone as they now sit above 18th-placed Everton on goal difference following the Toffees' goalless draw at Crystal Palace.
Smith will hope their victory over Lopetegui's side can serve as a springboard for a fine run in their battle at the bottom end of the table, with crunch clashes against Everton and Leeds United on the horizon.