Brighton coach Roberto de Zerbi's first game in charge could not have got off to a better start as two well-taken goals from Trossard gave the visitors a 2-0 lead by the 17 minute mark at a stunned Anfield.
Home supporters grew more frustrated as the hosts struggled to create openings, before Roberto Firmino's goal, which was initially ruled out for offside against Mohamed Salah only to be overturned by VAR, gave his side a 33rd-minute lifeline.
Brighton caused Liverpool more problems early in the second half, but their wastefulness at key moments proved costly as Firmino levelled with his fifth league goal of the campaign -- as many as he managed in the entirety of last season.
Liverpool completed the turnaround and appeared to be on course for a crucial victory after their slow start to the season when Brighton's Adam Webster put the ball in his own net in the 63rd minute.
But Trossard was not done yet, completing his treble seven minutes from time and ensuring Brighton left with the point they deserved, a draw that keeps them fourth in the standings, four points ahead of Liverpool in ninth.
"We have to fight through this," Liverpool boss Jurgen Klopp said. "The boys can play much better. My job is to create a situation where we can play much better and for that you need stability.
"We have to control the game, which is something we have been talking about for two or three years.
"You have to work step by step to get confidence back. We had confident moments today, but our passing game was not good today, we passed balls all over the place. We also can defend each goal much better."
Liverpool came into the match under pressure to get a busy October off to a positive start, with leaders Arsenal 12 points clear of Klopp's team prior to kick off against high-flying Brighton.
Their chances of bringing the Premier League trophy back to Anfield for the second time under Klopp looked to be a distant dream as Brighton took control in the Merseyside sunshine.
Firstly, Danny Welbeck flicked the ball through for Trossard to break the deadlock in the fourth minute - the ninth time Liverpool have conceded first in their last 11 league games - before the same man stroked home an almost identical second.
The goal out of nowhere that did get Liverpool back into the match was not celebrated with too much vigour, given the linesman's flag was raised, but replays showed Salah was in fact just onside when diverting the ball into Firmino's path.
Liverpool surprisingly had started the match with Diogo Jota, Darwin Nunez and Luis Diaz all on the bench, but the latter's introduction at the break immediately paid dividends, as the Colombia international supplied the pass for Firmino, whose superb touch and finish did the rest.
Liverpool took the lead in the 63rd minute with a goal Brighton goalkeeper Robert Sanchez will not want to see again, as he completely missed his punch, leaving Webster powerless as the ball rolled into his own net off his chest.
Brighton kept coming, with Welbeck denied by a brilliant save from Liverpool goalkeeper Alisson, before Trossard again arrived right on cue at the back post, becoming the third opposition player to score a Premier League treble at Anfield.
Information from Reuters was used in this report.