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.
Elsewhere in League One, Wycombe celebrated the New Year in style by climbing to the top of League One with a dramatic 2-1 victory over Exeter.
The match appeared to be heading for a draw after Luke Leahy and Tristan Crama exchanged goals, but the home side broke the deadlock with an Aaron Morley penalty in the fourth minute of added time.
It lifted them from third to first, replacing Birmingham at the summit after the Blues could only manage a 1-1 draw with Stockport.
Alfie May opened the scoring at Edgeley Park with a superb effort just outside the box in the sixth minute, but Chris Davies' side could not break through again despite further chances and MacauleySoutham-Hales levelled in the 78th minute.
Wycombe and Birmingham both have 50 points but Wanderers occupy top spot through a superior goal difference.
Harvey Knibbs struck twice as Reading defeated his former sid...
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.
Elsewhere in League One, Wycombe celebrated the New Year in style by climbing to the top of League One with a dramatic 2-1 victory over Exeter.
The match appeared to be heading for a draw after Luke Leahy and Tristan Crama exchanged goals, but the home side broke the deadlock with an Aaron Morley penalty in the fourth minute of added time.
It lifted them from third to first, replacing Birmingham at the summit after the Blues could only manage a 1-1 draw with Stockport.
Alfie May opened the scoring at Edgeley Park with a superb effort just outside the box in the sixth minute, but Chris Davies' side could not break through again despite further chances and MacauleySoutham-Hales levelled in the 78th minute.
Wycombe and Birmingham both have 50 points but Wanderers occupy top spot through a superior goal difference.
Harvey Knibbs struck twice as Reading defeated his former sid...
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Daily Post (Digital) - 1 Issue, January 02, 2025