In the end the hosts held on with 10 men and with backup goalkeeper Adrian on the field on what was a freezing afternoon in Merseyside.
Klopp said: “We scored two nice goals from set plays but the biggest chances we had were from open play. We could have and should have finished it off in that period, and we didn’t.
“In the second half it was hard work again and the boys put in another incredible shift.
“Of course life became more complicated with the red card and the change we had to make.
“We brought on a frozen goalkeeper. Everyone sitting here was not warm. Imagine going on there with shorts, a really thin shirt and gloves that are not made for keeping you warm.
“Then some people make the free-kick happen like that. We looked a bit silly in that moment but you have to accept it.
“We carried on fighting and Adrian helped us with two really good saves and crosses he caught, but with his cold feet he couldn’t kick the ball as far as he wanted.
“It kept it interesting but in the end the most important thing is we won the game. I am very pleased about that, happy about the effort of the boys and proud of the desire they showed. The red card made it a really special win.”
Liverpool’s victory equalled the club’s record of 31 top-flight games unbeaten, set in 1987/88. That run ended with defeat to rivals Everton, their opponents in their next game on Wednesday.
Alisson will now miss that derby through suspension.
Liverpool have also not kept a clean sheet in 12 matches in all competitions, but Klopp is concerned primarily about results.
He said: “None of these 13 games we have won was easy. We didn’t feel that for a second.
“It is not as though we feel ourselves as favourites in some games. We fight with all we have for the result.
“Results-wise it is an incredible period but we don’t think about the last 14 games, we have to think about the next one, which is Everton and a big one. We don’t think about the points gap.”
Brighton boss Graham Potter was pleased with the performance of his side but felt they paid the price for some lapses.
Potter said: “We put a lot into the game against a top team. I’m very proud of the effort and what we tried to do.
“We started OK but not good enough for how we really want to be. Two set plays made the scoreline 2-0 and you have to survive some moments.
“Over the course of the 90 minutes we did a lot well, showed some personality and some courage. Lots to be positive but still disappointed to come away with nothing.”