Chelsea looked comfortable after two far-post headers from Enzo Fernandez and Levi Colwill in the first half.
Facundo Buonanotte pulled a goal back with a curled finish and Gallagher was sent off just minutes later.
Mykhailo Mudryk won a second-half penalty after being brought down in the box by James Milner.
Some remarkable defending was key to keeping the scoreline intact with 10 men, despite Joao Pedro pulling another one back, and a late penalty scare.
Nizaar Kinsella assesses three key points which helped Chelsea earn the victory.
Land of the giants at the back
An all central defender back four of Colwill, Benoit Badiashile, Thiago Silva and Axel Disasi helped Chelsea dominate in both boxes.
Their size helped Chelsea open the scoring from two corners, with Badiashile setting up Fernandez's opener and Colwill nodding in a Nicolas Jackson flick-on.
Pochettino's calculation to harm an injury-hit Brighton, who were without Lewis Dunk and Adam Webster, by looking for set-pieces was key.
Silva led the way in an impressive second-half rearguard display to hold when down to 10 men for over 45 minutes.
Goalkeeper Robert Sanchez also put in one of his best displays yet against his former club.
Sterling and Mudryk perfect outlets
Raheem Sterling was outstanding throughout as Chelsea built their attacks, whether with 11 or 10 men.
He was both the key dribbler and the link man for a team without Cole Palmer, who had a slight injury.
Mudryk got a rare start as Palmer was benched and started slowly but when down to 10 men, his dribble down the middle earned Chelsea the vital penalty to win the match. Fernandez put it away, adding his second of the game..
Ill-discipline remains a problem
An ever-growing number of yellow cards marred some fantastic grit from Chelsea.
They have more cards than any other side in the Premier League, with 47 bookings and three reds now this season after Gallagher's dismissal.
Moises Caicedo was booked and risked a sending off while flying into further tackles after his initial caution but stayed on the pitch.
Brighton, who got caught up in the emotions of the game, also had several bookings. Both teams surrounded referee Craig Pawson after the match as VAR rightly ruled out a late penalty call against Colwill.