Murray: Netted the opener
Glenn Murray put Albion in front by heading in a Cox cross after 18 minutes, but Joel Porter levelled from close range on 75 minutes after keeper Michel Kuipers blocked a goal-bound shot from David Foley.
And midfielder Cox settled the issue with two minutes remaining by pouncing to score from close range after a pass by Murray.
Inconsistent Pools were hard done by not to have taken something from the game, and defender Michael Nelson should have put them ahead when putting a free header wide from six yards.
Brighton's big Dutch keeper Kuipers came to his side's rescue after only two minutes by pushing away a goal-bound 20-yard shot from Andy Monkhouse.
Steve Thomson was denied his first Brighton goal for the club through offside after firing home from 25 yards just before Murray opened the scoring, before later clearing a Porter effort off the line.