Bryson struck in the 58th minute with a powerful drive to give the visitors a five-point cushion over their opponents.
Kilmarnock did not have it all their own way, however, as they survived a bright start by the hosts that saw Marc Twaddle and Carl Finnigan waste decent chances.
Reduced admission prices ensured a larger than normal turn-out and created a vibrant atmosphere.
The hosts went close to breaking the deadlock as early as the second minute. Colin Healy’s corner was not dealt with by the Kilmarnock defence and Twaddle flashed a shot just wide of Cammy Bell’s post from 12 yards.
Towering striker Enoch Showunmi then got on the end of Twaddle’s lofted cross but the former Leeds player’s tame header bounced harmlessly off target.
In a bright start by Eddie May’s side, Carl Finnigan had a shot across goal palmed behind by the outstretched Bell after the marksman drove into the box.
Kilmarnock were struggling to get to grips with their opponents’ slick passing as the home outfit dominated the early exchanges.
However, Falkirk were dealt a blow when midfielder Vitor Lima limped off injured in the 18th minute and was replaced by Burton O’Brien.
The hosts were then nearly punished for a lapse in concentration at a short corner which culminated in Manuel Pascali’s close-range header being tipped over by Robert Olejnik from Bryson’s deep delivery in the 20th minute.
Kilmarnock were dealt an injury blow of their own on the half hour mark when captain Pascali was taken from the pitch on a stretcher following a challenge, with Jamie Hamill his replacement.
Falkirk were not able to maintain their high-tempo start to the game as Kilmarnock gradually exerted their influence on the bottom-of-the-table encounter.
May’s side enjoyed some success in floating long balls into Showunmi but were becoming predictable in their approach play as Killie pounced to clear the striker’s flick-ons.
Kilmarnock lacked ingenuity in the final third although Tim Clancy worked a one-two with Chris Maguire on the edge of the area before mis-hitting a poor shot goal ward that was easily picked up Olejnik.
The Bairns began the second half well, with Carl Finnigan heading wide from Showunmi’s cross.
Just seconds later, though, Falkirk goalkeeper Robert Olejnik was called upon to deny Liam Kelly from close range.
Bryson then powered forward and teed up Kelly inside the box but the midfielder hit his effort straight at the goalkeeper.
The Ayrshire side were proving more direct after the break and Olejnik made another good block from Maguire’s 18-yard stinging shot following Kevin Kyle’s flick-on.
Kyle and Falkirk defender Brian McLean were both able to continue following a clash of heads that required treatment from their respective physios.
McLean had a bigger headache in the 58th minute when Kilmarnock took the lead.
Maguire did brilliantly to hold the ball up at the corner before flicking a pass into the area for the supporting Bryson.
Olejnik got a hand to the midfielder’s low near-post shot but could not keep it out.
Falkirk manager May threw on Pedro Moutinho for Showunmi with 20 minutes left as the Bairns sought to salvage a point.
The players tried to stick to their passing game but it was not easy on a rugged surface.
Despite Falkirk pressing for an equaliser, Kilmarnock were next to register an attempt on target when Kyle rose high at the back post to meet Bryson’s free-kick but the Scotland cap’s header fell into the arms of Olejnik.
With Showunmi watching on from the bench, Falkirk had no late success in launching high balls into the area.
Kilmarnock utilised their final substitution with five minutes left by bringing on Conor Sammon for goalscorer Bryson.