The win is the Bees' first in six games and moves them one point above AFC Wimbledon, who play on Saturday, on goal difference.
Billy Kee had put Burton ahead from the penalty spot on 16 minutes after Barry Fuller fouled Jacques Maghoma, but Holmes grabbed an equaliser against the run of play as half-time approached - rounding Dean Lyness to score after a poor clearance.
Barnet looked a different side in the second half and went ahead after 53 minutes, Holmes sending Lyness the wrong way from the spot after Elliot Johnson's cross struck Cleveland Taylor's arm.
And the 25-year-old secured the win and his hat-trick after 66 minutes, firing home from just inside the area after a superb run down the right from Andy Yiadom.
Maghoma pulled one back from the spot in stoppage time, the third penalty of an eventful evening, but Barnet held firm under pressure to secure a vital three points.