Jake McCarthy briefly levelled the game for the visitors, but The Shaymen never looked anything other than comfortable and were good value for the win, which keeps their play-off hopes ticking over nicely.
With Billy Chadwick missing due to personal reasons, Jamie Allen started after his international duty with Montserrat, and won Town a penalty after 30 seconds latching onto Jeff King's lofted pass.
Hyde's penalty was saved by keeper Ethan Ross, but the Town striker converted the rebound for his seventh of the season.
The Shaymen went on to play some nice football, with Hyde and Jack Earing especially bright and purposeful in finding the ball and bringing others into play.
Weymouth hadn't really settled and were giving Halifax too much time on the ball, but Halifax were lacking composure in the final third.
Loan striker Andrew Dallas was lively and offered good movement for the visitors, but it came as a surprise to see the visitors equalise midway through the first-half given they had produced little to worry the hosts until then.
After conceding from two set-pieces against Sutton last time out, it was particularly disappointing to see it happen again as McCarthy nodded in Sean Shields' free-kick from close range.
There were shots on target by Earing and Kieran Green as Halifax offered the far greater threat and produced more quality on the ball.
Town had been sloppy at times but had got into good areas, and a few more precise passes could have seen them in on goal more often.
Town's experiecced heads at either end of the pitch - Hyde and Nathan Clarke - were stand out performers, Hyde for his link and hold up play and Clarke for his numerous headed clearances and general marshalling of the defence.
Bradbury, who scored in Town's victory against Weymouth in December, then produced an excellent header back across goal from a Town corner to restore their lead.
Town won the reverse fixture 5-1 earlier in the season but weren't finding it as easy to cut through their opponents this time against a more resilient Weymouth side, helped by the impressive Jacob Mensah at the heart of their defence.
But a crucial two-goal cushion was established when Hyde fired home left-footed from inside the box.
That put Town even more in control of the contest, but Ollie Harfield's shot from a tight angle that Sam Johnson kept out was a reminder that it was far from over.
Weymouth didn't give up but Town looked hungry for more, as Earing curled a shot just wide from the edge of the box and substitute Danny Williams fired off target.
Loanee Micah Obiero was handed his debut too - and hit the bar with two minutes remaining - as Halifax controlled the game and Weymouth waned as the clock ticked down.
McCarthy pulled one back in added time with a superb drive from outside the box, and there was a scare moments later when Dallas fired wide, but Town held on.
Halifax: Johnson, Byrne, Clarke, Bradbury, King (Obiero 70), Maher, Green, Earing (Spence 76), Senior (Williams 64), Allen, Hyde. Subs not used: Renshaw, Woods.
Scorers: Hyde (2, 47), Bradbury (31)
Shots on target: 7
Shots off target: 8
Weymouth: Ross, Brooks, Revan, Robinson, Wakefield (Thomson 66), McCarthy, Murray (Leslie-Smith 87), Harfield, Mensah, Dallas, Shields. Subs not used: Benfield, Ngalo, Fonkeu.
Scorer: McCarthy (23), (90)
Shots on target: 3
Shots off target: 5
Referee: Paul Marsden
Town man of the match: Hyde, Green and Clarke all contenders, but Jack Earing gets the nod for another classy performance in midfield, repeatedly finding pockets of space to operate in and more often than not causing Weymouth problems.