The 19-year-old from Sydney fired home from the edge of the penalty area in just the fourth minute to settle a lively encounter in east Lancashire.
Accrington huffed and puffed in the latter stages of the first half and throughout the second, threatening the visiting goal more and more as the tie boiled to a stirring finish.
But the staple requirement of a side trying to stave off the threat of a cup upset is to match the effort and desire of their lower division opponents.
And Leicester passed this test with flying colours as they were led superbly by their captain Stephen Clemence and veteran former Marseille defender Bruno N'Gotty.
They also had the upper hand in the quality stakes too, with strike trio Matt Fryatt, DJ Campbell and Iain Hume all looking lively in the opening stages.
In all fairness to Accrington, they were given their major headache very early in the contest when Wesolowski pounced on Stanley's failure to clear an N'Gotty cross that inadvertently hit the bar and found its way back out to the edge of the box.
And they tried admirably to force their way back into things with midfielder Jay Harris hitting the post before Andy Proctor saw his shot cleared off the line.
But their best chance of reply came when left winger Rommy Boco brilliantly controlled a loose ball before volleying towards the top right corner only to see Foxes keeper Paul Henderson tip the effort over.
Captain Peter Cavanagh also hit the post with a low right-footed curling free-kick with ten minutes left, but it just wasn't to be for the hosts.