The lively Jutkiewicz put his side ahead when he met Matt Lowton's right-wing cross and fired home at the near post.
Darikwa then teed up Spireites new boy Jake Orrell, who fired home to reduce the deficit at the break.
Daniel Jones threatened to level the game with a fierce effort on the hour mark, but it flew just inches over the top of goal.
However, Armand Gnanduillet did pull Chesterfield level in spectacular style when he converted a floated ball forward with an overhead kick that flew in off the post.
Jutkiewicz almost won the game late on for Burnley, but he was denied by Lee's diving save from close range, leaving Gnanduillet to wrap up the win in the dying seconds.body check tags ::