Song was sent off in the first half for elbowing Croatia striker Mario Mandzukic in the back, while Assou-Ekotto head-butted team-mate Benjamin Moukandjo in the 90th minute, as Cameroon's FIFA World Cup campaign fell apart.
Song and Assou-Ekotto's violent outbursts came on the back of rumours of unrest in the Cameroon camp with the players reportedly having threatened to boycott Brazil 2014 due to a pay dispute, while injured captain Samuel Eto'o was labelled a 'traitor' in the local press for a perceived disinterest in playing against Croatia.
With his team out of the running for a round of 16 berth, Finke apologised for their performance in Brazil and promised to investigate Song and Assou-Ekotto's behaviour.
"It's unimaginable," the German coach said after the game in Manaus.
"I saw this happen and I need to find out exactly what happened, why exactly these two players exploded.
"I hate to see that. It's not the image of Cameroon I want to project."
Croatia triumphed at the Arena Amazonia thanks to goals from Ivica Olic, Ivan Perisic and a brace from Mandzukic but Cameroon barely challenged the Europeans in arguably the worst performance of this World Cup.
"I apologise for the result, it really hurts," Finke said.
"We had no control on the game and I didn't like the behaviour of my team. We have to look forward - this team has players for the future."
Finke added that conceding three goals in the second half could not be accepted, despite playing over 50 minutes a man down.
"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," the 66-year-old said.
