The 26-year-old Australian midfielder last found the net in December 2011 but lashed home from eight yards to put the Latics firmly in command inside the first 10 minutes.
Danny Philliskirk had already put the visitors ahead in the opening minute and Preston, who made eight changes from the weekend, failed to stage a comeback, with Oldham tasting victory for the first time in six games.
It took the away side just 15 seconds to grab the lead when Philliskirk wrong-footed Thorsten Stuckmann and slotted the ball into the net.
And it was not long before the visitors were 2-0 up when Wesolowski finished emphatically from 10 yards after good work from James Dayton down the right.
It could have been three just two minutes later but North End goalkeeper Stuckmann recovered well to spare his own blushes after miscontrolling a back pass.
Joe Garner wasted Preston's best two chances to get back into the match while defender Tom Clarke also saw a header cleared off the line.