First half goals from Steven Milne and Ryan Conroy booked Dundee's place in the fifth round of the William Hill Scottish Cup at the expense of hosts Livingston.
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Milne put the SPL side ahead with a turn and shot nine minutes into an entertaining first half, while Conroy doubled their advantage from the penalty spot.
Livingston felt they should have had a spot kick of their own shortly afterwards, but despite dominating possession in the second half they were unable to net a reply.
The First Division side made one change to the team which lost for the first time in seven games last Saturday, Kyle Jacobs starting in place of Kevin McCann.
Visiting manager Barry Smith made two changes from the team which lost 3-1 at St Mirren last weekend, with Matt Lockwood and John Baird starting in place of Colin Nish, sent off in that game, and Nicky Riley.
Baird had the first chance of the game for Dundee, heading wide of the left-hand post after three minutes.
Livingston's Antony Andreu then had a shot from the edge of the box blocked, and at the other end, Baird sent an effort wide.
Dundee took the lead after nine minutes when Milne made a good turn inside the box and sent a low shot into the left-hand corner beyond the reach of home goalkeeper Andrew McNeil.
The SPL side almost doubled their lead five minutes later thanks to a good run up the right wing by Jim McAlister.
He crossed under pressure from the edge of box to Ryan Conroy, whose pacy diagonal shot went just narrowly wide of the right post.
Conroy succeeded in making it 2-0 from the penalty spot shortly afterwards, however, after Baird was brought down.
The Dundee forward stepped up for the spot kick and drilled a left-footed shot low into the net with McNeil going the wrong way.
Andreu tried to reduce the deficit with a turn and shot which Dundee goalkeeper Robert Douglas claimed easily.
Livingston were furious to be denied a penalty after 20 minutes when Liam Fox's shot from the edge of the box appeared to be handled on its way through to being headed off the line.
Milne had two more close-range chances as the game entered its second quarter, first having an effort headed off the line from a corner, then almost cashing in when McNeil could only parry Baird's raking 25-yard shot, the goalkeeper managing to recover and smother the ball before Milne could get it over the line.
Livingston worked their way into the opposition box on plenty of occasions but were always met by stout defending from the visitors, and saw a free-kick from Jesus Garcia Tena saved by Douglas.
Baird had a shot deflected out seven minutes from the break, but Dundee maintained their comfortable two-goal lead into the second half for which Martin Boyle replaced Milne.
Livingston dominated the possession as the second half got under way and saw Andreu test Douglas with an early turn and shot, while Fox and Baird traded speculative long-range efforts which went well over as the hour mark approached.
Burton O'Brien tried his luck with a free-kick from deep in midfield after Lockwood had fouled substitute Daniel Mullen, while Kyle Jacobs brought a good save out of Douglas with a powerful shot which the goalkeeper had to turn round his left post for a corner.
McAlister's shot into the turf was put out for a corner and the set-piece delivery from the right was saved by McNeil, while Iain Davidson's effort with the rebound was blocked inches from the line by Kyle Jacobs.
Livingston continued pressing and Marc McNulty headed over late on.
An even better opening for the same player right on 90 minutes saw him shoot straight into Douglas' arms.