Van Dijk hesitated under pressure from Arsenal's Gabriel Martinelli in the 67th minute, failing to deal with a long punt forward, then keeper Alisson rushed out and made no contact with the ball, leaving Martinelli a tap-in to make it 2-1.
It knocked the stuffing out of Klopp's league leaders and Leandro Trossard's stoppage-time goal confirmed their second league defeat of the season.
"We were growing more into the game, second half, we had our moments and then we concede that goal," Klopp said. "That doesn't help. It just shows the boys are human beings.
"Today we were really human."
Lacklustre Liverpool were nowhere near their best and their lead has now been cut to two points while champions Manchester City are five points behind with two games in hand.
"In general we can admit that Arsenal deserve the three points," Klopp said. "The circumstances were a bit strange.
"They scored goals and were many parts better than us. We have to play better football, that's clear."
Van Dijk, usually a colossus for Klopp, held up his hand and took full responsibility for his part in Martinelli's goal.
"Obviously these things don't happen too often in my career, but I will recover from this," he said.
"It is a physical game and I am not looking for excuses. I should have made a better decision - it hurts for me."
Arsenal had taken the lead early on through Bukayo Saka and were dominant but were deflated on the stroke of halftime when Gabriel's own goal gifted Liverpool an equaliser.
But they responded after half time, aided by Liverpool's defensive mishaps, and a third league win in a row leaves Mikel Arteta's side right back in the mix in the title race when defeat would have left them eight points adrift.
"It gives us momentum, three wins in a row, and the way we have done it we are back on it - really excited," Arteta said.
"It was a huge game that was going to dictate where we are and where we're going to be. What the players produced was phenomenal."
