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 after an hour with Hearts claiming offside The referee correctly over ruled his linesman and after allowing play to continue Easton connected strongly with the cross to head the ball down and past Hearts young keeper Craig Gordon and into the net.
Livingston finally made a cup-tie of it as they took the game to Hearts but couldn't quite engineer a second goal to force a replay.
The Livi supporters were magnificent in their attempts to encourage their team forward for the last 30mins of the game. Despite yet another loss, they clapped the team from the park, with the players returning the compliment.
So Hearts squeeze through into the Semi Finals. Surely they will be the team the others want to play in this seasons Scottish Cups penultimate round?