Nicky Forster twice went close for the home side with Mark Crossley pushing his shot round the post before the striker headed a Dean Cox cross over.
Craig Davies put the ball over the bar and Lewis Alessandra shot straight at the keeper as Brighton wasted chances.
But Murray earned the win when he took the ball neatly round keeper Crossley and slotted in from an acute angle.
Brighton manager Dean Wilkins told BBC Southern Counties Radio:
"I thought we were outstanding in the first half and could have gone in two goals up at the break.
I was a bit concerned that Oldham would come back at us and they did in the second half.
"But we withstood that and Glenn Murray scored one of the best goals you're likely to see at Withdean."
Brighton: Kuipers, Whing, El-Abd, Cox, Richards, Bowditch (Robinson 76), Thomson, Fraser (Martot 36), Elphick, Murray, Forster (Lynch 89).
Subs Not Used: Mayo, Hart.
Goals: Murray 61.
Oldham: Crossley, Eardley, Hazell, Stam, Allott, Jarrett (Lomax 76), Livermore, McDonald, Taylor, Davies (Smalley 27), Alessandra.
Subs Not Used: Wolfenden, Bell, Black.
Att: 4,815.
Ref: Mike Russell (Hertfordshire).
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