Gough bemused many by his decision to start with central defender Greg Strong up front and, by the time he realised his error, Livingston were 2-0 down and Hearts had the upper hand.
The visitors played much better in the second half but never looked like pulling the tie out of the fire.
Hearts moved in front inside the opening minute when Jamie McAllister curled in a free-kick and Lee Miller headed the ball over Colin Meldrum.
The second came in the tenth minute after great wing play from Saulius Mikoliunas set up McAllister who cracked home a stunning shot.
Shortly afterwards, Gough changed Livingston into a more familiar system with Strong dropping back and they looked a more solid outfit.
But, by then, Hearts were running the show and peppered Meldrum's goal with shots from all angles.
Mikoliunas set up a great chance for Miller but he swept his effort over the top when he should have done much better.
McAllister then came within inches of a third when Stephen Simmons set him up but he pulled his shot marginally wide.
Hearts had strong appeals for a penalty after half-an-hour when Gus Bahoken made a clumsy challenge on Miller but referee Alan Freeland saw nothing wrong.
Mark Burchill selfishly shot wide after more good work from McAllister when Hearts would surely have profited more had he played in Miller shortly before half-time.
Livingston made three changes at half-time in an attempt to salvage the game and Craig Easton pulled a goal back with a header after an hour with Hearts claiming offside.
Livingston finally made a cup-tie of it as they took the game to Hearts but Craig Gordon was seldom troubled and the home side moved into the last four.