But the Robins started the slower of the two sides and found themselves behind after Josh Morris’ strike inside five minutes.
The Bantams blew their chance to double their lead from the spot after 19 minutes when Billy Clarke’s sluggish penalty-kick was easily smothered by Lawrence Vigouroux on his debut.
But the home goalkeeper was almost left red-faced 10 minutes later when Clarke tried an audacious lob after robbing Southampton loanee Jordan Turnbull on the halfway line, only for his long-range effort to fall narrowly wide of goal.
The second half was a totally different affair as Swindon ran riot, scoring four goals inside 18 minutes.
Byrne levelled things up after 52 minutes when he managed find the net with a bending shot that evaded Bradford goalkeeper Ben Williams.
Kevin Stewart then slid through Byrne on the left flank who managed to poke the ball low past Williams, before completing his hat-trick after 68 minutes after dribbling past a host of Bantams defenders.
Town striker Jon Obika finished the rout after meeting a Turnbull through ball before slotting home.