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Just after the 30-minute mark, Yasin Ayari went clean through and levelled for the Seagulls with a low shot, before Dominik Szoboszlai's ball in from the right flew over Brighton goalkeepr Bart Verbruggen and into the net.
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With 20 minutes left, though, Kaoru Mitoma, pounced on a rebound to fire home.
And then with time running out, Brighton sub Hinshelwood scored with his first touch from Matt O'Riley's cross, with the goal given after a lengthy VAR check.
Express Sport rates the Liverpool players...
Liverpool player ratings vs Brighton (4-2-3-1)
Alisson: 6
Couldn't be blamed for the first goal - but appeared to be caught off guard by Grudan scuffing his finish.
However, slow to react for the Brighton opener and his influence going forward faded in second half.
Ibrahima Konate: 6
Solid in the air and his distribution out from the back was on point.
Was a calming influence on his side when Brighton tried to press late on.
Dominik Szoboszlai: 7
Did he mean it?
Frequently got on the ball and tried to create, and timed his run into the box well for his goal.
Likely to leave this summer, he struggled to make an impact and when he did get a sight of goal in the second half, he completely mishit his effort and was hooked after an hour.
SUBS
Luis Diaz (For Chiesa, 63) 6
Tried hard to get involved but his final ball was lacking
Curtis Jones (For Szoboszlai, 63) 5
Added energy but coughed up possesion a few times and wasted one big chance, pulling the ball behind Salah when his team-mate was well placed to score.
Darwin Nunez (for Gakpo, 63) 6
His physical presence briefly brought a new dimension to Liverpool's attacking play but never managed to get a clear opportunity on goal.
Wataru Endo (for Conor Bradley, 77) 6
Little time to make any real impact but was prepared to help his defenders in the closing stages.