The Reds only needed nine minutes to go ahead with Szoboszlai's cross-field pass superb flicked on by Salah into Bradley's path. The right-back then did excellently to get to the byeline and his pass was steered into the net by Elliott.
Bradley, looking to show what he can do with Trent Alexander-Arnold leaving the club in the summer and set to join Real Madrid, had a glorious chance to double the Reds' advantage but watched his side-footed shot go wide of the post after good work from Chiesa.
Alexander-Arnold also remained an unused substitute, as Slot opted to not have a repeat of the scenes in which the England international was booed by his own fans when he came on during the 2-2 draw with Arsenal.
But Brighton's desire for victory proved greater as Ayari finished off a fine move following Brajan Gruda's assist, although Szoboszlai's fortunate goal restored Liverpool's advantage.
The hosts were the better team after the break and credit must go to boss Fabian Hurzeler as his substitutions proved key, with Mitoma scoring within five minutes of his introduction and Hinshelwood netting 78 seconds after coming on.
That goal from Hinshelwood was the fourth quickest by a substitute in the Premier League this season after he came on with his 17-year-old cousin Harry Howell, who became the youngest Brighton player.
Salah only needs one goal or assist to match the Premier League-era record of 47 goal contributions in one season, currently held by Alan Shearer and Andrew Cole, although their totals came in 42-game campaigns.
With Liverpool leading 2-1, the Egyptian had a great chance to put the game beyond Brighton, only to shoot wide from eight yards when in a central position - and that miss proved crucial.
Salah will have one last chance to match the goal contributions record as Liverpool host FA Cup winners Crystal Palace on Sunday, when the Reds will be presented with the Premier League trophy.