The Seagulls, now without a win in four league games, found Pool goalkeeper Joe Lewis in excellent form as he pulled off a string of important saves to deny Sami Hyypia’s men, who were on top throughout.
Brighton, smarting from back-to-back away defeats, should have taken the lead after just 40 seconds.
Spanish striker Adrian Colunga found himself one-on-one with Lewis but scuffed his shot, allowing the on-loan Cardiff keeper to make an easy save.
Blackpool, forced to make a change at centre-back with Jeffrey Rentmeister replacing the unwell Peter Clarke, looked fragile in defence and Brighton’s Joao Teixeira saw a lot of the ball early on.
The gifted Portuguese playmaker, on loan from Liverpool, forced keeper Lewis to save low down after the Seagulls earned a free-kick and corner in quick succession.
Lewis showed alertness midway through the half by diving to confidently save an angled shot from the industrious Teixeira.
Blackpool, desperate for an improvement after suffering their worst start to a season in the club’s history, forced their first attempt on goal in the 27th minute.
A low shot by Polish midfielder Tomasz Cywka was well held by former Fulham keeper David Stockdale to deny the Seasiders their first goal in three matches.
A foul by Pool skipper Tony McMahon on striker Sam Baldock gave Brighton a free-kick just outside the area and a sweet left-foot strike from Joe Bennett flew into the side-netting.
Blackpool had a let-off when Teixeira superbly picked out Bruno Saltor on the edge of the six-yard box but the Spanish defender put the ball over when in a good position.
Keeper Lewis came to Blackpool’s rescue just two minutes after the break with an important save to deny Bruno at the far post following neat build-up play by Jake Forster-Caskey.
Lewis excelled himself again in the 50th minute to keep out a fierce goal-bound shot from Danny Holla as Brighton stepped up the pressure.
Colunga should have done better when Baldock unselfishly set him up 12 yards from goal but the Spaniard’s weak effort was blocked and the danger was cleared.
The hard-pressed Lewis denied Brighton again with a flying save to palm away a goal-bound free-kick from Holla just after the hour mark.
Brighton boss Hyypia made his first change by withdrawing Teixeira in favour of former Newcastle winger Kazenga LuaLua, and John Lundstram and McMahon both picked up bookings in quick succession for fouls on the pacey substitute.
Blackpool had a rare chance as the game opened up and former Nottingham Forest striker Ishmael Miller forced a smart save from keeper Stockdale with a shot from close range.
LuaLua later crossed cleverly to the far post where central defender Lewis Dunk headed narrowly over.
Blackpool, having used all their substitutes, had to battle the remaining six minutes and a further six of stoppage time with 10 men after Rentmeister suffered an ankle injury and had to be helped off.