Rutter, signed last week for £36million from Hoffenheim, was an unused substitute in Sunday's frustrating Premier League stalemate.
A disjointed match was crying out for Rutter to make himself an instant Elland Road hero, but Marsch resisted the temptation to turn to the France Under-21 striker.
Marsch said: "We thought about it. I could have done it. I just wanted to integrate him in the right way.
"I felt like the guys on the field, Willy (Gnonto) was close to pulling off a play, so I left him on, I thought Jack (Harrison's) performance was really strong.
"When you have new players you always want to set them up to succeed, so in the end I decided to wait maybe one more week before we unleash him."
Leeds created the better chances as their winless league run was extended to six matches, with Brentford goalkeeper David Raya making the game's decisive save from Gnonto's angled second-half effort.