A quickfire brace from Pascal Gross either side of half time sent the Seagulls on their way to the historic triumph before Tomer Hemed added a third to take the game beyond the previously unbeaten Baggies.
James Morrison did pull a goal back for the visitors, but it was too little too late as Brighton finally got off the mark in their fourth match of the season.
It was West Brom who were the first to threaten midway through the first half when summer signing Gareth Barry tried his luck from the edge of the box, only to see his strike fly narrowly over the crossbar.
Barry, who moved to within one game of Ryan Giggs's all-time Premier League appearance record this afternoon, came close again with six minutes remaining of the half, but this time his goalbound effort was well saved by Mathew Ryan.
Anthony Knockaert threatened for the hosts shortly afterwards, but the Seagulls' 34-year wait for a top-flight goal finally came to an end in first-half stoppage time when Gross wrote his name into the club's history books.
Brighton's first ever Premier League goal was a messy one, though, with Gross taking down Solly March's cross at the back post before cutting inside and squeezing his finish over the line despite slipping while shooting.
Having sent his side into the break with the lead Gross then doubled that advantage within three minutes of the restart, firing a low and accurate finish into the bottom corner from 25 yards.
The summer arrival from Germany turned provider shortly after the hour mark to well and truly wrap the game up, delivering a cross which allowed Hemed to get in front of his man and power a header past a helpless Ben Foster.
The Baggies, who had picked up seven points from their first three games, offered little in the way of a response as the clock ticked on, but they did pull one goal back with just over 10 minutes remaining when Morrison's volley crept over the line despite the best efforts of Shane Duffy.
The result sees Brighton climb up to 13th in the Premier League table, whereas West Brom drop to eighth after four matches.