Glenn Murray put the Seagulls 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 Cox settled the issue with two minutes remaining by scoring from close range after a pass by Murray.
Pools were hard done by and Michael Nelson should have put them ahead when putting a free header wide from six yards.
Kuipers came to Brighton’s rescue after only two minutes by pushing away a goal-bound 20-yard shot from Andy Monkhouse.
Steve Thomson was denied through offside after firing home from 25 yards just before Murray opened the scoring.