Alexander's 23rd minute penalty and Fletcher's ninth goal of the season had put the Premier League side firmly in charge at the break.
Substitute Dean Morgan pulled a goal back for the League One outfit in the 89th minute, but the Clarets survived a stoppage-time onslaught to bag their first win in 10 games.
Milton Keynes dominated the first half for long spells but they failed to turn their possession into clear-cut chances.
Robbie Blake blasted into the side-netting from 22 yards for the Clarets at the other end before Alexander's 23rd minute penalty put Burnley ahead against the run of play.
Alexander fired his spot-kick in off the post past the dive of Willy Gueret after Fletcher had robbed Dean Lewington of possession and been tripped in the box by Mathias Doumbe.
Blake then saw his goal-bound shot blocked by Luke Howell as the Premier League side upped their game, before Fletcher doubled their lead in the 35th minute.
Fletcher latched onto Chris Eagles' through-ball and coolly turned inside Doumbe before firing low past Gueret from 15 yards as the Dons appealed for offside.
The Dons batted back and Jason Puncheon's fierce long-range free-kick was well-held by Brian Jensen, who then spilled Danny Woodards' 25-yard half-volley wide.
Doumbe's close-range strike was blocked on the line by Richard Eckersley on the stroke of half-time after the Clarets defence failed to clear Puncheon's corner.
Burnley came roaring out of the blocks after the break and Eagles should have made it 3-0 but he fired against the bar from nine yards after Blake's deflected free-kick had spun kindly into his path.
Fletcher then headed against the post following a cross from Eagles, whose goal-bound shot was blocked on the line by Woodards.
The Dons piled forward late on and Peter Leven's 18-yard drive was well saved by the diving Jensen before substitute Luke Chadwick poked straight at the Burnley keeper from eight yards.
Morgan pulled a deserved goal back a minute from the end when he controlled fellow substitute Jude Stirling's cross on his chest and volleyed in via the underside of the bar from six yards but it wasn't enough despite a grandstand finish.