The duo, who both found the net against Doncaster Rovers seven days earlier, put the Imps into a 2-0 lead in the first half before Charlie Dawes responded for the Spireites.
Former Chesterfield youngster Burrow's goal came following good work by Ben Tomlinson on the quarter hour mark and after Tom Miller came close to adding a second, skipper Power curled home to double the advantage.
Chesterfield's consolation came immediately before they had an effort ruled out for offside when Luke Clarke followed up a Paul Farman save from a Romy Boco shot.
Amongst the substitutes was Frenchman Zak Belkouche, formerly of Stranraer, on the bench along with Jake Caprice and Luke Chambers and all three trialists were given a run out during the second half.
City: Farman (Chambers 66); Miller, Brown, Diagne, Newton (Caprice 45), Jordan, Nolan (Everington 75), Power (Belkouche 75), Tomlinson, Burrow (Simmons 82), Robinson (Kabba 45).
Chesterfield: Lee, Holmes (Darikwa h/t), Raglan (Evatt h/t), Cooper (Hird h/t), Humphreys (Jones h/t, Boco, Clarke (Ryan h/t), Featherstone (Morsy h/t), McElroy (Roberts h/t), Gnanduillet (Doyle h/t), Morgan (Dawes 13, Gardner h/t).
Attendance: 952 (148 visiting supporters)