Gassan Ahadme opened the scoring for Jimmy Floyd Hasselbaink’s side early on before Elié Youan equalised soon after.
Melkersen scored twice in ten minutes to put Hibs comfortably ahead until Ciaran Gilligan pulled one back for Burton.
Nohan Kenneh’s deflected effort late on ensured a comfortable scoreline in the end for Hibs, who were impressive against a tougher opponent.
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Lee Johnson once again picked two starting teams and changed things up from the first encounter against Hartlepool United on Wednesday.
Kevin Dabrowski started in goal with a back four of Lewis Miller, Ryan Porteous, Kyle McClelland, and Lewis Stevenson, with a midfield four of Jair Tavares, Kenneh, Joe Newell, and Aiden McGeady, and Christian Doidge partnering Youan up top.
The game started at a faster pace than the previous match, with Burton threatening early doors and the Brewers took the lead after just five minutes, Ahadme converting from close range after Louis Moult’s free kick had been deflected onto the post.
Hibs responded quickly, Kenneh picking out Joe Newell and the midfielder sent a perfectly-weighted ball through that Youan controlled before firing past Ben Garratt to level.
Hibs pushed for a second with the struggling to threaten despite the best efforts of Moult and Ahadme, but the Easter Road stood firm and Dabrowski was largely untested for the remainder of the half.
At the other end, McGeady had a couple of half-chances, one from a free kick that he curled just wide of the upright.
Hibs sprang out of the traps following the restart but it was Burton who had the first chance. Tom Hewlett picked out Gilligan and he forced a fine save from David Marshall.
Moments later Josh Campbell’s cross picked out Melkersen at the near post and he turned the ball past Callum Hawkins to put Hibs ahead.
Burton also made a raft of changes at the break but struggled to get out of their own half, and Melkersen made it two for him and three for Hibs when he was fed by compatriot Runar Hauge and finished beyond Hawkins from the edge of the area.
Hibs kept the pressure up and could have gone further ahead when Melkersen crashed a shot off the crossbar but the linesman’s flag was raised.
Jack Brydon, Rocky Bushiri, Chris Cadden, and Ryan Porteous all had half-chances from corners as Hibs turned the screw but it was Burton who struck next, Gilligan converting Williams Kokolo’s cross from close range following a quick break.
Kenneh restored Hibs’ two-goal lead ten minutes from the end when his effort from the edge of the area took a wicked deflection and flew in at the near post.
Hewlett then dragged a shot wide of Marshall’s left-hand post late on as the League One side chased another goal but Hibs held on for victory, Cadden going close late on with a driven effort that just cleared the bar.