Max Cleworth's late strike was not enough to stop Phil Parkinson's side from dropping out of the top two.
Wrexham had the ball in the net within the opening moments, but George Dobson was flagged for offside.
It was the home side who broke the deadlock with 11 minutes played.
A crossfield ball from Phillips found Keillor-Dunn, who curled in his 10th goal of the campaign.
The Tykes doubled their lead midway through the first half when Phillips struck from outside the area after a mistake from Eoghan O'Connell.
After the break, Corey O'Keeffe produced a fine goal-line block to deny substitute Steven Fletcher.
Wrexham pushed but Ben Killip produced numerous saves from close range.
Cleworth set up a tense finale as he smashed home after a goalmouth scramble with 10 minutes to play, but Barnsley held on.
This story is from the January 02, 2025 edition of Daily Post.
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This story is from the January 02, 2025 edition of Daily Post.
Start your 7-day Magzter GOLD free trial to access thousands of curated premium stories, and 9,000+ magazines and newspapers.
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