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.
John Baird had the first chance of the game for Dundee, heading wide of the left-hand post after three minutes.
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.
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 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.
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.