Price, 21, scored in both halves as the Baggies ran out 3-2 winners to consign the Red Dragons to back-to-back league defeats following last season's rise from League One - their third promotion in a row.
It was a carnival atmosphere inside the Racecourse Ground but Wrexham could not use that to their advantage as Price put West Brom in front after 20 minutes. The Everton academy graduate peeled away from the hosts' defence superbly to latch onto Tom Fellows' cross before firing home from close range.
And things soon went from bad to worse for Phil Parkinson's side when Josh Windass, a summer signing following his departure from beleaguered Sheffield Wednesday, had to be replaced with a suspected hamstring injury as the visitors grew in confidence.
However after Kieffer Moore was allowed to wander freely into the right-hand channel, the 6ft 5in striker's cross eventually found its way through to Lewis O'Brien, who slotted home Wrexham's first league goal of the season on home soil.
Having weathered the storm, Wrexham started the second half the better of the two teams and almost took the lead when club-record signing Nathan Broadhead fired just wide following another dangerous cross from Moore.
But West Brom started to grow into the game following the introduction of Jed Wallace, and the winger soon fired Ryan Mason's men back in front after getting on the end of a lovely pass from fellow substitute Michael Johnston.
And West Brom soon made it 3-1 just seven minutes later when Price headed home from Johnston's cross, leading to the travelling fans to start chanting 'Reynolds, what's the score?' in the direction of the watching Wrexham co-chairman and Hollywood star, Ryan.
Sam Smith set up a nervy finish when he headed past West Brom goalkeeper Josh Griffiths in the fifth minute of stoppage time, but it was too little too late in the end for Wrexham.