The hosts took the lead under fortuitous circumstances after just 13 minutes when Joel Byrom's curling free kick took a big deflection off the head of Stuart Beavon that sent it past Mark Oxley in the Oldham goal.
Davies stepped up to take the spot kick himself and placed his effort into the top corner to claim his first goal for the club.
The hosts could have killed the game off 10 minutes after the restart, but Scott Laird put his shot wide when he had plenty of time to pick his spot.
Davies almost doubled his personal tally when he curled a low effort just wide, while Rooney squandered a good chance at the other end when he was caught in two minds as to what to do with a header.
Oldham did claw themselves back into the match with just over five minutes remaining when Tarkowski powered a header in at the near post, but it was too little too late for the visitors.
The result leaves Preston sixth in the League One table while Oldham remain in 14th place.