Half-back Chris Atkin was the hero for the Lions, kicking a drop goal four minutes from the end of normal time to break a 12-12 stalemate.
But Richard Marshall’s side, who missed a gilt edged chance to close the gap on fourth-placed Toulouse after the French side lost at Dewsbury, will look back on two costly errors around the 66th minute, when they led 12-10.
A late challenge on Atkin by Brandon Moore gave the Lions half back a penalty chance, which he managed to steer wide of the posts, only for Fax centre Steve Tyrer’s 20 metre restart to go out on the full, giving Atkin a second opportunity he was never going to waste.
And when Shane Grady put the ball down on the 30 metre line, Swinton positioned Atkin for the decisive one point.
Atkin put Oliver Davies over for a converted try, and when Moore was sin-binned, the hosts increaded their interval lead to 10-0 with a Shaun Robinson try.
Fax stormed out of the blocks after the break, Moore redeeming himself with some smart work around the ruck, the hooker finally creating a try for Mitch Cahalane, Tyrer converting.
When Wood cut back against the defence from Sharp’s pass to score and Tyrer added the conversion, Fax were ahead at 12-10. But that was just the start of the drama.
Swinton: Murphy; Robinson, Hankinson, Lloyd, Butt; White, Atkin; Bracek, Waterworth, Barnes, Paisley, Davies, Field. Substitutes: Carberry, Nicholson, Bate, Barlow
Halifax: Sharp; McGrath, Heaton, Tyrer, Williams; Murrell, Johnston; Cahalane, Moore, Murray, Grady, Barber, Grix. Substitutes: Morris, Walne, Douglas, Wood
Referee: C. Straw (RFL).