Kevin Mirallas justified his recall with the goal that broke the deadlock early in the second half
Everton won away from home for the first time since September as they deepened the gloom for the Champions with a hard-fought victory at the King Power Stadium.
Kevin Mirallas broke a tight contest open six minutes into the second half when he raced onto a long ball down the middle by Joel Robles and slotted past Kasper Schmeichel.
Then, with the clock ticking into time added on for stoppages, Romelu Lukaku out-muscled Wes Morgan by the touchline and powered towards goal before tucking the ball into the corner of the goal to wrap up a big win for Ronald Koeman's team.
The manager had rung the changes for this Boxing Day trip to Leicester hoping to bounce back from last Monday's defeat in the Merseyside derby, deploying a back three that included Mason Holgate, drafting Gareth Barry and Joel Robles in for the injured James McCarthy and Maarten Stekelenburg and dropping Ross Barkley back to the bench.
The Blues started well but quickly found their hosts difficult to break down and by the mid-way point of the first half, Leicester had assumed control of the game even they hadn't tested Robles beyond a couple of efforts from distance by Amatey and Demarai Gray.
Everton's attacking threat had been limited to angled balls aimed for Lukaku, one of which almost broke for him in the 29th minute, until Ramiro Funes Mori headed over following a corner late in the first half and Leighton Baines raked a dangerous ball across the area that just eluded Mirallas.
The first period had offered few clues as to how the visitors could break the Foxes down but it took a momentary lapse in concentration by Leicester's centre halves as Robles sent an accurate ball forward for Mirallas to profit and benefit from a slight deflection from Marcin Wasilewski to help the ball past Schmeichel's out-stretched leg and make it 1-0.
Ranieri responded with a double substitution that introduced Riyad Marhez and Leonardo Ulloa around the hour mark and the Argentine almost levelled soon after his introduction. Danny Simpson swung a cross in that found Ulloa unmarked in front of goal but he planted his header straight at Robles.
Koeman, meanwhile, withdrew Barry in favour of Tom Davies and then Barkley for Mirallas and after the 18-year-old had tried his luck with a couple of efforts from 20 yards, the Blues' best move of the second half almost made it 2-0 with seven minutes to go.
Barkley released Lukaku down the right and he centred to meet Idrissa Gueye's run but with the goal at his mercy, the Senegalese international could only scoop it over the crossbar as he slid in.
The killer second goal did arrive in injury time though with Leicester desperately trying to repeat the salvation act they pulled off at Stoke last time out. Lukaku came away from a challenge with Morgan with the ball and a trademark run and shimmy to outfox the last defender later, he had the space he needed to plant the ball past the stranded ‘keeper and send the travelling Blues away in good Christmas cheer.