Michel Kuipers did well to keep out a low shot from midfielder Joe Colbeck in the 26th minute before Andrew Crofts headed an Elliott Bennett corner narrowly over.
Oldham striker Abbott curled a shot over shortly after the break following a defensive slip by Brighton skipper Adam Virgo.
But Abbott broke the deadlock 12 minutes after the interval with a deflected shot from 20 yards which left goalkeeper Kuipers wrong footed. Abbott's effort lacked pace but Kuipers could not recover after the ball came off Virgo.
Veteran striker Nicky Forster was twice off target for Brighton before defender Hazell's first goal of the season clinched the points for Oldham nine minutes from time.
Hazell beat Kuipers with a side-foot finish as Brighton failed to deal with a free-kick into the six-yard box by Irishman Alan Sheehan. Welsh midfielder Andrew Crofts headed against the bar for the struggling Seagulls in stoppage time.