A third-round shock looked possible when powerful striker Uche Ikpeazu put the League Two side ahead on 25 minutes but the visitors, who are fifth in the Championship, rallied with a strong second-half performance.
Northern Ireland international Stuart Dallas headed the equaliser from an Alex Mowatt cross on 56 minutes and Mowatt grabbed the winner, flicking on Pontus Jansson's header on 63 minutes.
In the fourth round, Leeds, who won the competition in 1972, will travel to face the winners of non-league side Sutton United and AFC Wimbledon.
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(Reporting by Neil Robinson, editing by Pritha Sarkar)
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