Leicester made it two wins from two Coca-Cola Championship matches under new owner Milan Mandaric as John McGreal's own goal extended Burnley's worst winless run for more than quarter of a century.
The Foxes, who began the Mandaric era in style with a comprehensive 3-0 victory over Midlands rivals Coventry on Saturday, ended their recent dismal record against Burnley when McGreal poked the ball past Danny Coyne in the 67th-minute.
The goal stretched Burnley's run to 13 matches without a win - the club's worst streak since a 24-match sequence in 1979.
Leicester had lost the previous three meetings between the sides 1-0 and had not scored against the Clarets in the last five matches since Trevor Benjamin scored in a 2-1 win at Turf Moor in March 2003.
But Rob Kelly's side started brightly and created the best opportunities of the first half but were unable to break the deadlock.
The Foxes, who scored three times inside the opening 26 minutes against Coventry on Saturday, went close through Alan Maybury's 20-yard shot after three minutes.
Burnley responded with an effort from top scorer Andy Gray but his header sailed harmlessly wide of Paul Henderson's left-hand post.
Ade Akinbiyi forced Henderson into the only save with a low shot from inside the area midway through the half - but it was textbook stuff for the Australian.
Gareth McAuley should have put Leicester in front when he got on the end of Mark Yeates' corner but the defender put his header wide from six yards.
Burnley might have done better with headed chances of their own but James O'Connor and Steve Caldwell failed to hit the target.
Akinbiyi got another shot on goal at the end of the half but again the effort did not extend Henderson.
Caldwell retained the Burnley captaincy despite Wayne Thomas returning to the side at the expense of Michael Duff.
Clarets boss Steve Cotterill made just one other change to the side which lost against Wolves on Saturday with Chris McCann replacing Steve Jones.
Kelly stuck with the same starting XI that got the Milan Mandaric era off to a victorious beginning against Coventry. The only change to City's 16-man squad saw Stephen Hughes replace Danny Cadamarteri on the bench.
There were signs at the end of the first half that Burnley were beginning to take control of the game and after the interval they picked up where they had left off.
Caldwell flashed an angled drive across the face of goal before McAuley threw himself in front of a goalbound volley from the same player soon after.
Iain Hume forced Coyne into his first save of the match just before the hour mark and that signalled the start of some concerted Leicester pressure which would eventually bring the winning goal.
Patrick Kisnorbo headed Levi Porter's corner narrowly wide in the 63rd-minute and Elvis Hammond forced a fine one-handed save from Coyne with a fierce 12-yard shot just seconds later.
Three minutes later McGreal put through his own net.
Burnley had chances to equalise but a series of blocks from Kisnorbo to twice deny Kyle Lafferty and then McGreal ensured the points for Leicester.
Teams:
Burnley Coyne, Thomas, Caldwell, McGreal,Harley (Lafferty 70), Elliott (Jones 66), James O'Connor,Gudjonsson (Djemba-Djemba 70), McCann, Gray, Akinbiyi.
Subs Not Used: Duff, Garreth O'Connor.
Booked: Caldwell.
Leicester Henderson, Maybury, Kisnorbo, McAuley, Johansson,Yeates (Hughes 46), Jarrett, Johnson, Porter (Stearman 82),Hume (Hammond 60), Horsfield.
Subs Not Used: Logan, Kenton.
Booked: Kisnorbo, Hughes.
Goals: McGreal 67 og.
Att: 10,274
Ref: M Dean (Wirral).