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The hosts fell behind in the 19th minute, with Eoin Doyle scoring on his first start for the club.
However, Michael Brown ensured that lead only lasted eight minute, as he rifled home from 35 yards to level the scores at one apiece.
Young striker Dom Poleon then completed the turnaround in the 31st minute.
The League Two side threw everything at Brian McDermott's men in the second half as they looked for a route back into the game.
It was not to be, however, as Leeds weathered the storm to put their name in the hat for the second-round draw.