Everton suffered their fifth top-flight defeat this season despite taking the lead against Arsene Wenger's side through Wayne Rooney in the first half.
The Toffees crumbled after Idrissa Gueye was shown a second yellow card in the 68th minute, with the heavy defeat adding to the pressure on Koeman. However, the Dutchman called for calm.
"We played against the top five from last season but this is not the position on the table that is good for Everton and I understand [that]," Koeman told Sky Sports.
"I've been managing a long time and don't tell me what can happen. I don't think it's too late.
"It's not a time to start to be crazy I'm responsible for the team. We lost 5-2 today, it's a big result even against teams like Arsenal. We will see what will happen.
"I told the players I believe in them, in their commitment … The decision is not in my hands."
The former Southampton boss pointed to Gueye's dismissal as the turning point, claiming Everton still had a chance with 11 men on the pitch.
"Of course, the final result is really very disappointing," he said. "At 2-1 we stayed in the game. OK, I need to say Arsenal was the better team, had the better chances, but at 2-1 you can get a point.
"After the sending off of Gana it was really difficult. It brought more space to Arsenal and then honestly they have the qualities to break you down and that happened and that's disappointing.
"I think we had a good start in the game. We fought for everything in the first half but sometimes it's impossible to stop everything, [Jordan] Pickford stopped a lot of balls in the first half but that's football.
"They have such high-quality players and it's difficult. After the sending off it's more difficult to stay in the game."