Ryan Williams grabbed the only goal of the game with a deflected shot before Albion ended the game with nine men after an injury to substitute Alex MacDonald and the dismissal of defender Shane Cansdell-Sherriff.
The hosts should have been ahead in the sixth minute but the unmarked Adam McGurk drove Chris Hussey's cross against the legs of goalkeeper Barry Roche.
Andrew Fleming hammered the Shrimps' first chance wide before McGurk forced Roche to fist a well-struck free-kick to safety.
Morecambe almost snatched the lead in the 24th minute when Jack Sampson turned Kevin Ellison's cross against the foot of a post from a tight angle.
Ellison played a key role as the visitors went ahead in the 34rd minute, pouncing on a defensive error and then seeing his shot deflect off team-mate Williams and past wrong-footed keeper Jordan Pickford.
Pickford denied Morecambe a two-goal lead with a superb one-handed save to keep out a Fleming shot in the 53rd minute.
Hussey volleyed wide from MacDonald's cross as Burton pressed for the equaliser.
Cansdell-Sherriff was sent off in added time after picking up a second yellow card - one of nine dished out by referee Tim Robinson.