Two goals from Roberto Firmino, an Emre Can header, a Philippe Coutinho effort and an own goal from Lewis Dunk saw Klopp's side return to Merseyside with a victory that kept them three points behind third-place Chelsea.
Klopp was confronted by Albion boss Hughton on the full-time whistle, initially thought to be a result of the Reds boss' exuberant celebrations on the touchline throughout the game.
Both managers refused to talk about the incident in their post-match interviews but Klopp has now lifted the lid, claiming there was a mix-up over a handshake.
“No one asked me about it after the game when I could have killed the discussion immediately. I have to apologise – not that I did something wrong on purpose, it just happened.
“I went to go to Chris to shake his hand but Mo Salah passed my way so I gave him a hug and Chris wasn’t too happy about that.
"We shook hands and he said what he thought – and it is what I think as well – that you should shake the hand of the other manager first, but it was not on purpose and after the game I spoke to Chris and his assistant.
"I think and I hope everything is fine. For one or two times in my life I was disrespectful but I would never be disrespectful in a moment like this.
"I can only apologise, it was not my intention.”
Liverpool remain fourth in the Premier League rankings, while Brighton occupy 12th place after 15 fixtures.