But the Latics were left fuming after ref Nigel Miller waved away a strong spot-kick appeal of their own.
When John Thompson crashed through the back of club record £140,000 signing Joe Garner from a Paul Arnison cross three minutes before the break, Graham stepped up and showed a cool head to send veteran keeper Mark Crossley the wrong way from the spot.
It gave the Cumbrians caretaker boss Greg Abbott his second win in five days following the shock dismissal of Neil McDonald.
Carlisle had the ball in the net in the fifth minute when Jeff Smith headed in from close range, but he was flagged offside.
And chances were at a premium throughout the game. Carlisle almost added a second in the 58th minute when Arnison whipped in a dangerous low cross that just evaded the in-sliding Garner.
Just two minutes later they broke at pace from a corner with Marc Bridge-Wilkinson unselfishly playing in Kevin Gall, who dragged his shot horribly wide from 18 yards with just the keeper to beat.
Oldham began to turn the screw and the Blues were lucky to escape a strong penalty appeal in the 73rd minute as Craig Davies went down under a Danny Livesey challenge in the box, but the ref waved their angry protests away.
And Davies went close three minutes later as he smashed a low drive across keeper Keiren Westwood and inches wide of the far post.
The Latics should have levelled in stoppage time when Livesey's desperate clearance struck Westwood and rebounded to Stuart Giddings, who had his head in his hands as he smashed well over the bar from ten yards with the goal at his mercy.