The home side took the lead in the 20th minute when Simon Adingra picked the pocket of former Brighton midfielder Alexis Mac Allister and fired home, but Salah levelled in the 40th minute, ending a flowing move with a trademark left-foot finish.
The Egyptian gave his side the lead from the penalty spot in first-half stoppage time after Dominik Szoboszlai was pulled down by Pascal Gross, and Salah struck his shot to the left of keeper Bart Verbruggen.
Substitute Ryan Gravenberch, who came on for Harvey Elliott at halftime, missed a golden chance to put Liverpool two up in the 54th minute when his close-range effort from Szoboszlai’s pass cannoned back off the crossbar.
Dutchman Gravenberch, who signed for Liverpool from Bayern Munich on the last day of the transfer window, made another fine chance in the 65th minute, but his shot lacked power and was easily saved.
Brighton had their chances in the second half and they howled for a penalty in the 69th minute as Kaoru Mitoma cut in from the right and unleashed a shot which they claimed hit the arm of Virgil van Dijk, but the referee disagreed.
Brighton forced an equaliser in the 78th minute when Andy Robertson decided against trying to clear Solli March’s free kick, allowing defender Dunk to volley home.
If Gravenberch’s miss was bad, João Pedro‘s for Brighton in the 84th minute was arguably worse as he failed to hit the target from close range with the goal at his mercy, but the hosts held on for the draw to stay sixth in the table on 16 points, one behind third-placed Liverpool.
“I think we were the better side throughout and then we had a funny five minutes,” England international Dunk said. “We have got to be better than that. We have shown great character to come back and get the draw.”
Liverpool captain Van Dijk said his side were disappointed not to take all three points.
“We came here to win. I think we had opportunities to do so and in the end a draw, after a difficult game. Quite open at times. Quite even but looking at our team, we had opportunities to win,” he said.