Trossard's double in the opening 17 minutes put the visitors in command in head coach Roberto De Zerbi's first game in charge.
Liverpool hit back through Roberto Firmino before the break and the Brazil striker equalised early in the second half of a pulsating Premier League encounter.
Adam Webster's own goal put Liverpool ahead for the first time, but Trossard completed his treble with seven minutes of normal time remaining to give Brighton a deserved point.
Trossard gave De Zerbi a dream start to his reign, taking a clever flick from Danny Welbeck and drilling home with his left foot from inside the area after beating Trent Alexander-Arnold.
Alisson made fine saves from Welbeck and Trossard but was powerless to prevent the latter doubling his tally with a clinical left-foot finish after Alexander-Arnold's error.
Robert Sanchez thwarted Mohamed Salah, but the Egypt forward's touch gave Firmino the opportunity to find the back of an empty net, with the goal awarded after a video assistant referee (VAR) check for offside.
Firmino showed great composure to equalise on the break nine minutes into the second half, taking a pass from half-time substitute Luis Díaz and producing some trickery to round Lewis Dunk before applying the finish.
The turnaround looked complete when Sanchez paid the price for flapping at Alexander-Arnold's corner, with the unfortunate Webster turning into his own net.
Yet Brighton continued to pose a huge threat and, after Alisson denied Welbeck, Trossard sealed his hat-trick by superbly converting a left-wing cross.
Alexander-Arnold forced an excellent save from Sanchez in the closing stages, but Liverpool was unable to find a winner.