Van Dijk had fired the Reds into a 2-0 first half lead with two headers at Anfield but Alisson was given his marching orders in the 76th minute for handling the ball outside the box.
Lewis Dunk scored from the resulting free-kick for the visitors to set up a tense finale for the home side on Merseyside.
However, Jurgen Klopp’s men held on to claim all three points and move 11 points clear at the top of the table.
Asked about Alisson’s sending off in his post-match interview with BBC Sport, Van Dijk said: “We should have done better. We put him in that situation but it is what it is. We have to deal with it. It is a disappointment but we go again.”
The Liverpool FC defender continued: “They [Brighton] are a good side. They play football and take risks. 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. Unfortunately we got that red card with Alisson then we just had to defend.”
Liverpool FC have only dropped two points so far this season and they currently find themselves as the heavy favourites to win the title this term.
Defending champions Manchester City were held to a 2-2 draw by Newcastle United earlier in the day.
The Reds will return to Premier League action on Wednesday night when they welcome their Merseyside rivals Everton to Anfield.
Liverpool FC will then take on Bournemouth away from home next Saturday lunchtime.
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