After Martin Paterson and Chris McCann both went close inside the opening seven minutes, it was no surprise when Burnley took a deserved lead as Chris Eagles sent Wade Elliott racing to the byline and his cross was superbly steered into the far corner of the net by the head of McCann.
Chris O'Grady had Oldham's first opportunity of the evening but, after holding off a challenge from Clarke Carlisle, pulled his shot well wide of the target.
O'Grady showed his strength again, this time withstanding a tackle from Christian Kalvenes but missed the target with his strike.
Oldham came close to an equaliser when Kevin Maher's free-kick just evaded the head of central defender Reuben Hazell.
McCann should have notched for a second time when he played a one-two with Paterson but placed his shot wide.
The Clarets increased their lead as a surging burst by Eagles ended with his shot being parried by Greg Fleming only for Paterson to convert the rebound.
Oldham boss John Sheridan made a triple substitution in a bid to get back into the match and Burnley goalkeeper Brian Jensen was called upon to make saves from Danny Whitaker and Lee Hughes.
Paterson secured Burnley's passage into the third round of the competition with a fine left-foot finish after the Oldham defence failed to deal with a long clearance from Jensen.
Hughes should have pulled back a consolation goal with seven minutes remaining but was denied by Jensen when clean through.