Celtic increased their lead at the top of the Scottish Premier League to seven points with a win at Dens Park.
The visitors' first-half domination only resulted in one goal, an excellent Georgios Samaras overhead kick from 10 yards.
Dundee were lively after the break, with Fraser Forster clawing Colin Nish's header off the line.
But a classy bit of control and lob by Gary Hooper, after Lustig's long ball, sealed victory over bottom club Dundee.
Celtic went into this one looking to extend the gap between themselves and nearest challengers Inverness and Motherwell and get their fifth league win on the trot.
Dundee's aim was to narrow the concerning nine point gap between themselves at the foot of the table and Ross County in 11th.
It was an encouraging start for the home support at Dens Park, then, to see Stephen O'Donnell fire in the first real shot on target.
The midfielder benefited from poor control by Victor Wanyama on the edge of the box, but his effort went over the bar.
After 10 minutes Celtic began to assert their authority and Charlie Mulgrew's corner kicks were their weapon of choice.
Efe Ambrose got his knee on the first of Mulgrew's left-footed deliveries, but the unorthodox attempt came back off the bar and Wanyama drove the rebound over.
However, another one of Mulgrew's set-pieces was the catalyst in the visitors taking a 16th minute lead.
Again, it was hit left-footed from the right side and Tony Watt's side-foot volley from a tight angle at the back post forced Rab Douglas into a close-range save.
But that block saw the ball loop to Samaras, whose expert overhead kick from 10 yards found the roof of the net despite a desperate attempt by Jim McAlister to head off the line - a fine way for the Greek to score his eighth goal of the season.
The striker could have made it nine for the season shortly after when Beram Kayal put him through at the corner of the box, but Douglas touched his curling shot round the far post to prevent Celtic doubling their lead.
Dundee struggled to gain a foot-hold in the game throughout the rest of the first half and manager Barry Smith would have been pleased the deficit remained at one goal.
Ambrose did threaten once more just before referee Iain Brines blew for the interval, surging from midfield to arrow a right-footed shot Douglas's way from 25 yards, but the keeper was equal to it.
Celtic re-emerged for the second half with Lustig replacing Mulgrew, who had damaged a rib.
Perhaps it was the relief at Mulgrew's absence that boosted Dundee at the beginning of the second half.
McAlister linked intelligently with Nish on the right before sending in a shot which Wanyama deflected wide.
Ryan Conroy could not force anything from the resultant corner, but he was soon scampering down the left creating havoc.
His excellent cross was met by Nish, whose near-post header seemed destined to ripple the back of the net before the hand of Forster clawed it away - easily the home side's best chance.
Spurred on, John Baird then went very close again with a shot from the edge of the box on 55 minutes which Emilio Izaguirre slid in desperately to divert away from goal.
Lustig soon showed his attacking prowess by swinging in a low ball which came off Kyle Benedictus and required excellent reactions from Douglas.
The Swedish right-back then set-up Hooper to double Celtic's lead on 71 minutes.
Lustig's long-ball should have been dealt with, but Hooper raced clear of Benedictus and Declan Gallagher to show an exquisite first touch and lob over Douglas from just inside the box.
Almost immediately, another Nish header forced Forster to tip over to deny Dundee a quick reply.
Scott Brown, ever lively, angled a pass to set Hooper up again on 80 minutes, but the Englishman narrowly missed the target.
Douglas did well to deny Hooper again late-on and despite a couple of Dundee corners causing some confusion in the box, Celtic made it to the final whistle with a clean sheet.