Paulo Dybala, Romelu Lukaku, Gianluca Mancini and Bryan Cristante scored the goals for the hosts, who proved to be far too strong for their Premier League opponents on the night.
Brighton now have a mountain to climb ahead of the second leg of their last-16 clash on March 14 if they are to advance to the final eight of the European competition.
Roma were firmly on the front foot in the early exchanges, with Brighton goalkeeper Jason Steele reacting to keep out a powerful header from Lukaku in the third minute, before Dybala missed the target with an effort from a promising position less than a minute later.
Lukaku again tried his luck in the fifth minute, with the forward striking from the edge of the box, but his effort was always moving wide of the post.
Steele was called into action for a second time in the match in eighth minute, with the goalkeeper keeping out a strike from Stephan El Shaarawy, who had broken clear following a Brighton corner.
Roma made the breakthrough in the 13th minute of the contest through Dybala, who beat the offside trap before rounding Steele and finishing into the back of the net.
Dybala's effort was initially disallowed for offside, but the decision was overturned following a VAR check, with the Argentine therefore celebrating for the second time in quick succession.
Danny Welbeck almost answered back for Brighton in the 25th minute, but the forward's header from a Simon Adingra cross was excellently saved by Roma goalkeeper Mile Svilar, before Lukaku headed wide down the other end during an open period of the contest.
Roma managed to double their advantage just before the half-time break, though, with Lukaku breaking into the penalty box following a mistake from Lewis Dunk and then impressively finishing past Steele.
Brighton had another opportunity in the closing stages of the first half, but Svilar was again on hand to keep out an attempt from Welbeck, with the hosts leading by two goals at the interval.
Steele made another smart save to keep out a header from Lukaku early in the second period, before Svilar did well to deny Adingra down the other end, with both teams having their moments.
Welbeck again missed the target after smart work from Adingra in the 57th minute, and Brighton were made to pay in the 64th minute, with Mancini converting a cross from El Shaarawy to hand Roma a three-goal advantage in the last-16 contest.
It went from bad to worse for Brighton in the 68th minute, with Cristante heading a delivery from El Shaarawy into the back of the net to make it 4-0 to the home side.
Dunk just missed the target for the visitors in the 81st minute, with the Seagulls looking for a consolation, but Roma managed to secure a clean sheet, and the Serie A outfit are now the overwhelming favourites to advance to the quarter-finals of the competition.