Scott Phelan's 90th-minute equaliser looked to have given the Shaymen the chance of an extra-time tilt at the visitors.
But in truth, their hopes were really dashed by a poor display from Jim Vince's side who took a step backward from the previous week's league win over Skelmersdale.
The desire and application shown against the Merseysiders was missing.
The defending was poor, the finishing patchy and in between the passing was not too clever either.
And that all added up to a disappointing afternoon for the majority of the near-700 crowd inside the Shay.
Town were almost handed the perfect start when, after five minutes, Colin Daniel drove the ball into the Curzon area.
It bounced kindly for Ashley Stott on the left of the area but twice he had shots blocked by the visitors' keeper David Carnell.
Within a minute, midfielder James Ogoo spotted Jon Kennedy off his line and tried his luck from 40 yards, but the Town keeper got back to tip the ball over the bar.
Peter Moore and Daniel combined well in the 16th minute to play the ball in for Jonathan Smith whose shot was kicked away by the keeper.
And moments later Stott found Kris King but his driven shot from the left went across the face of goal and wide.
Curzon started to get a foothold in the game after that and Kennedy was forced to dive full length to his right to pluck an Alex Elliott free kick out of the air.
And Lincoln Adams had to time his challenge on Norton inside the area to perfection as the striker looked to get on the end of an Elliott pass.
Adams also cleared the resulting corner, but Town were finding it hard to get anything going forward and there was a scare when Norton went down in the box under a challenge from Moore, the referee deciding there was nothing untoward in the incident.
Norton curled a free kick round the post from 20 yards before Town had one final first half chance as King swept the ball from left to right.
Moore nodded it into the path of Smith who drilled in a shot that the keeper claimed at the second attempt under pressure from Daniel and Stott.
The second half could hardly have started any worse for Town as Kris King sent a poor back pass towards the edge of his own area and Norton nipped in and fired past Kennedy.
The Shaymen should have been level within three minutes as Moore crossed in from the right and the ball reached Stott, but he fell over with the goal gaping.
Within 60 seconds Smith picked up the ball 30 yards from goal and struck a beautiful shot towards the top corner only for Carnell to move across his line and save.Ogoo's 55th minute shot from the edge of the area found its way into the Town net only for the referee to spot an infringement by Norton in the area.
And Nigel Jemson had to be at his best moments later to block another Norton shot.
Daniel gave the home fans a lift on the hour when he won the ball wide on the right close to halfway and surged towards the edge of the area before firing in a shot that Carnell fumbled round his post.
But Kennedy had to be alert again soon after to keep out Michael O'Connor's well struck shot.
Moore fired over the top before Daniel was denied by the keeper again, Carnell tipping a left foot shot over the bar.
Stott headed a Jemson cross into the keeper's arms and Chris Worsley fired narrowly wide as the sides continued to look for goals.
And Town thought they had at least earned themselves extra time when Moore played in a low ball for Phelan 15 yards from goal and he turned to fire home.
Kennedy saved well to deny Elliott moments later, but he was powerless to prevent what proved to be the winner in the second added minute, Norton pouncing again, this time after Luke Hinsley - King's replacement - had failed to deal with a hopeful Curzon through ball.


STAR SHAYMAN
Few players caught the eye on a disappointingly flat afternoon at the Shay.
Ironically, loan star Colin Daniel did with some direct running at the Curzon Ashton defence and nifty work around the edge of the area when he got a sight of goal.
Jon Kennedy helped keep the Shaymen in the game with a couple of great saves.
And there was no way he could have been faulted for either of the Curzon goals which came as a result of defensive lapses.
Lincoln Adams is also exempt from the criticism thrown at the defenders after another solid display at the heart of the defence
Daniel gets the three points on offer in our season-long competition, although he may not be here for much longer, while Kennedy picks up two and Adams one.

MATCH FACTS
FC Halifax Town: Kennedy, King (Hinsley 53), Moore, Adams, Meadowcroft, Jemson, Phelan, Baker, Smith (Ellis 53), Stott, Daniel, Subs (not used): Ellison, Coo, Allen.
Booked: Daniel (foul 71)
Sent-off: None
Goals Phelan 90

Curzon Ashton: Carnell, Russell, Whelan, Birch, Jones, Ogoo (Booty 78), Elliott, Worsley, Norton, Byrne (Edghill 73), O'Connor. Subs (not used): Gill.
Booked: Byrne (foul 68), Worsley (foul 69)
Sent-off: None
Goals: Norton 48, 90

Attendance: 697
Referee: S Hunter.

POINTS
Jonathan Smith 13, Peter Moore 10, Junior Brown 9, Paul Gedman 8, Colin Daniel 6, Jon Kennedy 6, Scott Phelan 5, Ashley Stott 5, Adam Morning 4, Steve Payne 3, Lincoln Adams 4, Cavell Coo 3, Nigel Jemson 2, Craig Smith 2, Justin Walker 2, Damian Allen 2, Phil Senior 1, Luke Hinsley 1.

YOU SAY
two defenders cost us the game against a pub team, u were not interested from start - The turnbull