The Indomitable Lions were reduced to 10 men in the first half when Alex Song struck Mario Mandzukic in the back with his elbow, and went on to lose 4-0 as the Croatians poured forward.
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But faced with calls to quit from the Cameroonian press, Finke did his best not to be drawn further into the increasingly messy situation.
"It's not for me to decide. It has nothing to do with me. Please, give me a night's rest," he said.
"I want to present another side of this team against Brazil, this is about discipline, we have to play 11 versus 11."
He later added to Fifa.com: "It's a tough result to take. It's always difficult to play with 10 men, but that's still no excuse for tonight's performance.
"It wasn't enough for me as coach. Everybody has to do something more when it's 10 versus 11. After today, we must be clear to prepare for our last game against Brazil.
"Sorry for this result. The style of defending and poor behaviour from the red card is why we gave up four goals.
"There's no doubt that Croatia were more clinical in front of goal, but Cameroon also had chances to score. We need to move on and look to the future, because this team is full of promising players."
Finke, a German, became the third coach of Cameroon's World Cup qualifying campaign back in May 2013, and is currently working alongside four local assistants appointed by the country's Football Association to ensure a local influence on the technical staff.
The idea was mooted by Cameroon legend and World Cup 1990 hero Roger Milla, who hit out at Finke before the tournament, insisting he has no faith in the former Freiburg boss, and once again called for him to go in the build-up to Wednesday's game.