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.
Gavin Tomlin's second-half brace fired Southend to a 3-0 win against Chesterfield at Roots Hall.
The striker took his tally to 11 goals in 10 games with two fine efforts after Ryan Cresswell had opened the scoring.
The Shrimpers were easily able to make it 13 matches unbeaten as they moved up to fourth place in League Two.
Defender Cresswell put Southend in front in the 51st minute when he volleyed home his eighth goal of the season from 12 yards after Marc Laird's shot from Michael Timlin's cross had been blocked.
Paul Sturrock's side doubled their lead 10 minutes later when Tomlin swapped passes with Kevan Hurst before charging in to the box and heading home the midfielder's right-wing cross from 12 yards.
Tomlin was at it again in the 69th minute with the best goal of the night as he nodded in a chipped left-wing cross from Chris Barker following a number of successive passes.
And Rochdale denied Cheltenham the chance to move up to second in League Two with a thumping 4-1 win at Spotland.
The hosts had lost their last four games but there was little to suggest a fifth would be on the cards against their high-flying visitors.
Rochdale created the clearer chances in an open first half. Bobby Grant, who passed a late fitness test on a shoulder injury, fired a fierce drive against the crossbar while Scott Brown was at his best with a double save to deny Ashley Grimes and Andrew Tutte in quick succession.
Grant opened the scoring in the 38th minute, collecting a short pass from Dele Adebola on the left edge of the penalty area before sending a stunning curling effort past Brown.
Grimes made it two-nil with an equally impressive strike from 25 yards in injury time at the end of the first half, his polished finish leaving Brown rooted to the spot.
The visitors were briefly back in the game when Phil Edwards scooped the ball into his own net three minutes into the second half after Josh Lillis had parried Chris Zebroski's effort.
But another fine strike from Grimes in the 67th minute, quickly followed by a tap-in from Adebola, made the result safe for the hosts.