New head coach Marsch's side served up 19 attempts at goal in Saturday's clash at the King Power Stadium but Foxes winger Harvey Barnes bagged the only goal of the game seven minutes after the hour.
Defeat in the lunchtime kick-off kept fifth-bottom Leeds just two points clear of the drop zone but third-bottom Burnley then blew the chance to close the gap when falling to a 4-0 defeat at home to Chelsea on Saturday afternoon although the Clarets still have one game in hand on the Whites.
Elsewhere, 15th-placed Brentford gave themselves a huge boost in their bid to beat the drop, the previously out-of-form Bees providing a fresh twist in the relegation battle by recording a 3-1 success at bottom-of-the-table Norwich City.
Improving Newcastle United, who are 14th, further boosted their tally via a 2-1 win at home to Brighton.
Leeds remain one point ahead of fourth-bottom Everton but Frank Lampard's Toffees have three games in hand and are in action on Monday evening at Tottenham Hotspur.
Second-bottom Watford are at home to Arsenal on Sunday afternoon.
Based on the latest odds for relegation following Marsch's first game in charge, this is where Leeds are now predicted to finish - odds for the title determining where the top sides are expected to end up.