The Pilgrims’ experimental XI competed well before half-time but were put on the back foot when Cook cracked home a beauty just past the hour mark.
United were killed off eight minutes from time when substitute Akurang looped home a neat header.
Jason Lee had opted to shuffle his selection options, with established stars Gareth Jelleyman, Ian Ross and Marc Newsham starting on the bench, while Spencer Weir-Daley and Tom Ward were left at home nursing injuries.
The home side, fancied to be play-off contenders in Blue Square Bet South, looked strong from the outset and striker Craig Calver drove too high from distance, while Matt Wilson scuffed an effort well wide for the Pilgrims in response.
Kern Miller rose highest to plant a header over the bar following Ben Milnes’ corner, while City replied with Max Cornhill heading marginally wide from Ishmael Welsh’s left-wing cross.
Korey Dyer-Stewart — handed an extended opportunity to impress in attack — fired in an effort which deflected a fraction wide off Mark Haines.
City stepped up a gear after the break and stand-in skipper Nathan Stainfield had to produce a tremendous block to deny Jamie Slabber from close-range.
The Pilgrims should have taken the lead on the hour mark — but found themselves trailing a minute later. Mark Jones flicked on Dan Haystead’s long kick and Newsham, on moments earlier for an injured Dyer-Stewart, uncharacteristically slammed the ball wide.
The Clarets took full advantage and took the lead when COOK finished with real aplomb from 25 yards, giving Haystead no chance.
Ross was introduced and fired two free-kicks off target, while Haystead produced a top save to keep out a Calver drive.
City wrapped up their victory on 82 minutes with AKURANG looping a header over Haystead into the far corner following Jordan Parkes’ free-kick assist.
United almost reduced the deficit in the final minute, but Stuart Searle produced an instinctive save to grab Stainfield’s header on the line following Ross’ corner.
Chelmsford (4-4-2): Stuart Searle; Justin Miller (Ricky Brindley 64), Mark Haines, Kenny Clark (Nicholas Bondon 30), Aiden Palmer; Anthony Cook (Craig Parker 76), David Bridges (Joel Ledgister 76), Max Cornhill (Dave Rainford 83), Ishmael Welsh; Craig Calver (Jordan Parkes 76), Jamie Slabber (Cliff Akurang 64). Sub (not used): Tom Warren (gk).
Boston (4-4-2): Dan Haystead; Conor Marshall, Kern Miller, Nathan Stainfield, Gary Silk; Ben Fairclough (Jason Stokes 78), Ben Milnes (Ian Ross 67), Matt Wilson (James Reed 89), Kallum Smith (Joe French 82); Mark Jones, Korey Dyer-Stewart (Marc Newsham 56). Subs (not used): Jason Field, Gareth Jelleyman, Travis Munn, Ricky Drury (gk).