Alisson came out of his area as Leandro Trossard chased a long ball downfield and was adjudged to have handled as he touched it away from the Belgian.
Dunk should have pulled a goal back eight minutes before the interval during Brighton's best spell, but the defender clumsily shinned the ball wide from Dan Burn's cross, having perhaps been put off by Yves Bissouma failing to get fully out of the way from a clearly offside position.
"We brought on a frozen goalkeeper", said Klopp. Imagine going on there with shorts, a really thin shirt and gloves that are not made for keeping you warm.
'Some people let the free kick happen like that. "It's not easy. We should have gone better than we did in the second half but you have to give them credit as well". He starred in the 3-2 victory at West Ham in Mourinho's first game in charge, scored the goal that started Spurs' recovery from 2-0 down in the 4-2 win over Olympiakos in the Champions League, and yesterday scored twice in a 3-2 win over Bournemouth. Their keeper made some good saves.
Still, thanks to Van Dijk, who is in contention for the Ballon d'Or award in Paris tomorrow, Liverpool had a sufficient cushion to survive.
Hours after City's shaky defence conceded an 88th-minute equaliser in a 2-2 draw at Newcastle, van Dijk ventured forward to power home two headed goals from set pieces and lead Liverpool to a 2-1 win over Brighton at Anfield.
Brighton goalkeeper Mat Ryan denied Sadio Mane with a fine reflex save but was powerless when Van Dijk made his run to meet Alexander-Arnold's corner at the near post.
This time it was from a left-wing corner as Alexander-Arnold delivered a ideal cross and the Dutch defender rose magnificently to beat Dunk and power a bullet header past a helpless Mat Ryan.
Liverpool have also not kept a clean sheet in 12 matches in all competitions, but Klopp is concerned primarily about results.
Alisson will now miss that derby through suspension.
Trent Alexander-Arnold floated in the flawless free-kick and van Dijk rose powerfully above Adam Webster to head in his second goal of the season and first since the opening day of the season. I am very pleased about that, happy about the effort of the boys and proud of the desire they showed. "We don't think about the points gap".
"My aims are to win everything we play for", the Brazil worldwide told Sky Sports.
Chelsea's grip on fourth place suddenly looks a little less tight as they suffered a first home defeat by West Ham United for seven years, Aaron Cresswell's goal sealing a 1-0 win for the Hammers who had lost five of their last six games.
"We don't have to think about these kind of things".