Ollie Watkins sent Villa into a ninth-minute lead before James Maddison levelled, but the home side were back ahead in the 32nd minute due to an own goal from Harry Souttar.
Leicester scored twice in the latter stages of the first half through Kelechi Iheanacho and Tete to turn the game around, before Dennis Praet made sure of the points for the Foxes in the 79th minute.
The result has moved Leicester into 13th position in the Premier League table, while Villa sit 11th, seven points above their opponents on Saturday afternoon.
Villa took the lead in the ninth minute of the contest through Watkins, with the striker on hand to convert from close range after Emiliano Buendia had crashed one against the crossbar.
Leicester were level just three minutes later, though, with Maddison converting from close range after a pass from Iheanacho, who had taken the ball away from Boubacar Kamara.
Villa were back ahead in the 32nd minute when Watkins's shot from the edge of the box hit Leicester debutant Souttar before finding its way into the back of the net.
The game's fourth goal arrived in the 41st minute, though, with Iheanacho heading a Harvey Barnes cross into the back of the net to level the scores at Villa Park.
Leicester then took the lead in the closing stages of the first period, with debutant Tete rounding Emiliano Martinez before finding the back of the net, with the goal capping an incredible opening 45 minutes.
The Foxes scored with their only three shots on target in the first half, and it was some turnaround from Brendan Rodgers's side after twice falling behind to the home side.
Villa had the first half-chance of the second period, with Leon Bailey striking from distance, but Danny Ward was on hand to make the save between the Leicester sticks.
Both sides struggled to create openings in the period that followed, but Leicester scored their fourth in the 79th minute through Praet, who rounded Martinez before finishing, with Alex Moreno failing to cut out a pass from Barnes.
The win is Leicester's first in the Premier League since November 12, while the defeat is Villa's first in England's top flight since Boxing Day against Liverpool.
Unai Emery's side now face a trip to the champions Manchester City next Sunday, while Leicester will be bidding to build on the victory when they welcome Tottenham Hotspur next Saturday.