Well claimed the ball had crossed the line when home keeper David Mitchell collided with a post while clutching an over-hit Scott McDonald cross.
Cammy Kerr set up Craig Wighton then followed up to score after keeper Craig Samson blocked the striker's shot.
Marcus Haber side-footed home the second from striker Wighton's cut-back.
A second win in a row lifts Paul Hartley's side off the bottom of the Scottish Premiership table ahead of Ross County, who host Rangers on Sunday, and level with the visitors and Hamilton Accies.
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The ghost goal will haunt the match officials all weekend.
On the right flank, striker McDonald looped in a cross that was careering towards the top corner.
Mitchell began to furiously back-pedal as it became clear the ball was hurtling towards the back of the net.
The goalkeeper grabbed at the ball, but its movement, plus his momentum, carried him and the ball a good yard over the line.
There was amazement when the officials then decided no goal had been scored.
Motherwell - players, backroom staff and fans - were incensed, and no wonder. A crucial moment that had a huge bearing on this match as Dundee took control.
Wighton has been a hotly tipped young prospect for some time. Against Well, he was excellent.
There was a lot of huff and puff from Dundee in the first-half, but that little bit of quality in the final third came from the jinking feet of the 19-year-old.
The striker has imagination, skill, awareness and a directness that frightens defenders.
He created both goals. For the first, his effort was saved before Cammy Kerr fired home the rebound.
For the second, Wighton showed skill, strength and drive before cutting back for Haber to score his first Dundee goal.
Wighton almost capped his superb display with a great run and shot that was well saved by Samson. He deserved a goal for his all-round display.
A lot has been said about how tight the Premiership is this season, as long as runaway leaders Celtic are taken out of the equation.
That was the case here for large parts, although the ghost goal incident was pivotal in this game.
Dundee were the better side after taking the lead, but Motherwell were the better side for most of the first-half.
They created several chances - McDonald had a close-range header saved, Lionel Ainsworth had a corner tipped on to the bar and a shot blocked.
The ghost goal changed this game. From there, Dundee took their chance, got their goal, took control and - crucially - got themselves off the bottom of the table for a while at least.
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Dundee manager Paul Hartley: "The first home win of the season. I felt we deserved that today.
"We might have got a wee bit of a break with Motherwell's so-called goal, I haven't really seen it, but it's the breaks that we have not had.
"I felt we were comfortable today in the way that we played. Second-half, I thought we were excellent and we deserved that today.
"Craig Wighton has been around and about the squad from a young age, from 15, a lot of expectation put on his shoulders, but we have tried to ease him in there at times.
"But now we feel has matured, he has got stronger and you have seen in his performance today and even last week there is a different side to him now in terms of on the ball but even off the ball in his defensive work.
"Craig has got such outstanding ability and talent and that is the standard he has set himself now. I think it's his time now."
Motherwell manager Mark McGhee: "I have seen the video again, I saw it at the time, I could see clearly. The boy's feet are over the line, the balls ahead of him.
"There is absolutely no doubt it is a goal. It is an absolutely shocking decision.
"I just asked [the referee] for an explanation. Of course they don't speak to you.
"I'm as angry with my own team's first-half performance as I am with the referee's decision, or the linesman's decision.
"We made poor decisions, we never did the things we spoke about - getting down the sides of their back three. We conceded a goal.
"Goals do change games and the goal we might have scored at a time when they were really feeling under pressure - we needed that goal."
Match ends, Dundee 2, Motherwell 0.
Second Half ends, Dundee 2, Motherwell 0.
Foul by Paul McGowan (Dundee).
Dom Thomas (Motherwell) wins a free kick on the left wing.
Danny Williams (Dundee) is shown the yellow card for dangerous play.
Cameron Kerr (Dundee) wins a free kick on the right wing.
Foul by Ross MacLean (Motherwell).
Corner,  Motherwell. Conceded by Danny Williams.
Attempt saved. Craig Wighton (Dundee) right footed shot from outside the box is saved in the top right corner.
Attempt saved. Dom Thomas (Motherwell) header from very close range is saved in the centre of the goal.
Attempt saved. Chris Cadden (Motherwell) right footed shot from the left side of the box is saved in the top right corner.
Attempt missed. Stephen McManus (Motherwell) right footed shot from the centre of the box is too high.
Corner,  Motherwell. Conceded by Cameron Kerr.
Goal!  Dundee 2, Motherwell 0. Marcus Haber (Dundee) right footed shot from the left side of the six yard box to the bottom right corner. Assisted by Craig Wighton.
Foul by Kevin Holt (Dundee).
Scott McDonald (Motherwell) wins a free kick in the defensive half.
Darren O'Dea (Dundee) wins a free kick in the defensive half.
Foul by Scott McDonald (Motherwell).
Danny Williams (Dundee) wins a free kick in the attacking half.
Foul by Louis Moult (Motherwell).
Foul by Tom Hateley (Dundee).
Richard Tait (Motherwell) wins a free kick in the attacking half.
Attempt missed. Craig Wighton (Dundee) right footed shot from outside the box misses to the right.
Attempt missed. Kevin Holt (Dundee) left footed shot from the centre of the box is close, but misses to the right.
Corner,  Dundee. Conceded by Stephen McManus.
Foul by James Vincent (Dundee).
Richard Tait (Motherwell) wins a free kick in the defensive half.
Cameron Kerr (Dundee) wins a free kick on the left wing.
Foul by Dom Thomas (Motherwell).
Attempt missed. Craig Wighton (Dundee) header from the centre of the box is close, but misses to the right.
Corner,  Dundee. Conceded by Stephen McManus.
Foul by Danny Williams (Dundee).
Ross MacLean (Motherwell) wins a free kick on the left wing.
Substitution, Motherwell. Ross MacLean replaces Keith Lasley.
Substitution, Motherwell. Dom Thomas replaces Lionel Ainsworth.
Substitution, Dundee. Danny Williams replaces Kevin Gomis.
Stephen McManus (Motherwell) is shown the yellow card for a bad foul.
Kevin Gomis (Dundee) wins a free kick in the defensive half.
Foul by Stephen McManus (Motherwell).
Attempt missed. Kevin Holt (Dundee) right footed shot from outside the box is high and wide to the left.
Craig Wighton was the star of the show against Motherwell as Dundee secured their first home win of the season, but it was one tinged with controversy.