Marsch's future is in doubt after the Whites lost at Elland Road, leaving them in the bottom three of the league.
Bobby Decordova-Reid and Willian scored in the second half to give the Cottagers a two goal cushion.
The chants against Marsch and the club's board were heard in the last 10 minutes of the game.
The home side had taken the lead in the 20th minute and were on the verge of their first win since August.
They gave up another close-range goal six minutes later, this one from a corner.
The Whites looked vulnerable and huffed and puffed in their attack all day.
Illan Meslier made two great saves in the first half, the first one stopping Harrison Reed from scoring the opener and the second one stopping Andreas Pereira from tying the game.
Decordova-Reid and Willian were the ones who took their chances and made sure that the result wouldn't have changed.
It gives the Cottagers a second win in a row and leaves Marsch to ponder his future.
This felt like a must-win game for the American after Thursday's loss atLeicester left them out of the bottom three.
It was only a week ago that they dominated the league leaders at Elland Road and Marsch was eager to take advantage of this.
In a bright and bold opening 20 minutes, backed by a vociferous crowd, they were able to open the scoring after a blocked shot by Jack Harrison.
This would be as good as it got for the home side, who lost their way as soon as the game started.
The home side was in danger all the time, needing to rescue them at both 1-0 and 2-0.
Willian tapped home to put daylight between the sides after Decordova-Reid glanced in his hammer blow of a headed ball.
There is a powder keg waiting to ignite in fury at the club's failures, like the one that happened on a smaller scale on Thursday.
It came to pass with chants against the board and the manager, the latter getting the lion's share.
At the end of the game, Marsch took to the pitch to applaud the fans. It could be the last place the American is seen.
This is a big feather in the cap for the team. Their fifth win of the season was the same amount as their entire last top-flight campaign. The win puts them in the top seven.
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