Roared on by 8,000 fans Thomas Tuchel’s side dominated and were rewarded after the break with Antonio Rudiger bundling in their opener before Jorginho rolled home a penalty.
Timo Werner twice had goals ruled out before the break while Leicester, who looked flat after Saturday’s celebrations, offered little threat until Kelechi Iheanacho gave the visitors a lifeline with a cool finish in the 74th minute.
It was a nervy climax and Ayoze Perez wasted a glorious late chance to equalise but Chelsea clung on for a priceless win that moved them into third place on 67 points with one game remaining, meaning a win at Aston Villa on Sunday will guarantee them Champions League football next season.
For Leicester there is now the looming possibility that they will let a Champions League berth slip through their fingers for the second season in succession.
They could have ensured a top-four finish had they won at Stamford Bridge but will now drop to fifth, on goal difference, if Liverpool win at Burnley on Wednesday.
Leicester, who have 66 points, finish with a home game against Tottenham Hotspur while Liverpool’s final game of the season is at home to Crystal Palace.