Liverpool broke the deadlock after 18 minutes, Van Dijk heading in Trent Alexander-Arnold's free-kick.
The pair combined for the second midway through the first half, this time the defender heading in Alexander-Arnold's corner.
Brighton threatened more after the break and were given hope when Alisson saw red for handling outside the area.
From the subsequent free-kick, Lewis Dunk caught substitute goalkeeper Adrian cold to make it 2-1 with 11 minutes remaining.
But Liverpool held on to equal their longest unbeaten run of 31 top-flight matches.
Brighton drop to 15th with 15 points.
Liverpool: 4 Dec v Everton (H)
Brighton: 5 Dec v Arsenal (A)