Doughty, who suffered a gashed knee in the first half, played through the pain barrier and helped set up both goals as the Shaymen came back from 2-0 down late on.
The defender, who played with a pronounced limp throughout most of the match, put Martin Foster clear down the left on 76 minutes and it led to a penalty when Denny Ingram was shoved over in a goalmouth melee that followed Foster's cross.
Craig Midgley stepped up to take the spot-kick and sent Daggers keeper Tony Roberts the wrong way by dummying to his right before sending the ball low, into the bottom left corner.
Six minutes later, Doughty fired over a centre from the left touchline that Foster headed back across the face of goal for substitute Neil Ross to blast Town level from about ten yards.
Halifax could have won it in the last minute, but Steve Haslam's superb long-range effort after an unchallenged run goalwards, curled just wide of the left-hand post.
Earlier, Daggers had looked at set to register their fifth win from six games thanks to Chris Moore's first-half double.
Striker Moore tucked a low shot under keeper Ian Dunbavin from Glen Southam's pass on 15 minutes and then put the visitors two up following a defensive blunder.
Town midfielder Steve Bushell played a back pass to Dunbavin that got held up on the rain-soaked pitched and Moore nipped in to block the keeper's attempted clearance and tap in the rebound.
Former Brentford trainee Moore has now scored five goals in four games and has hit 13 for the season, as Daggers look to secure a play-off spot.
The visitors were languishing in the lower reaches of the Conference table, but a run of fine results have lifted them towards the top five.