Poole struck after just seven minutes, burying a low first-time shot from eight yards after Antony Sweeney had headed down a left-wing cross.
Poole wasted a gilt-edged chance to double the visitors’ lead three minutes later before Hartlepool goalkeeper Scott Flinders did well to deny Luca Scapuzzi.
Oldham stopper Alex Cisak made a brilliant save by touching Sweeney’s fierce drive on to the crossbar and dominant Hartlepool, for whom Micky Barron is in temporary charge, went close again through Andy Monkhouse.
Paul Dickov’s Latics improved after the break, but Tom Adeyemi shot inches wide after being set up by Filipe Morais.
Gary Liddle should have added to Hartlepool’s tally, only to miss the target when clean through on goal, while Oldham’s final chance saw Flinders make a comfortable save from Dean Furman in time added on.