Two strikes in the opening ten minutes including 32-year-old Cureton's excellent strike, were enough to kill off miserable Barnsley.
The Tykes created a chance with the game less than a minute old when the ball broke to Marc Richards who produced an overhead kick that drifted wide.
Colchester were handed the lead on a plate after only three minutes when a comedy mix-up in the home defence gifted them the opener.
Ex-Barnsley defender Chris Barker aimed a pinpoint cross from the left for striking giant Chris Iwelumo to head down and Tykes defender Adam Eckersley crazily side-footed the ball to stunned keeper Nick Colgan.
He succeeded in prodding the ball invitingly for Karl Duguid who had the simple task of slotting a shot past the hapless Colgan.
Colchester, without an away win since early December, ensured another three points would be heading to Essex after just ten minutes with another goal that came as a result of some clueless defending by the South Yorkshire outfit.
Duguid started the move feeding a crisp pass for Kevin Watson who cut back a clever ball for Cureton who drove a sharp shot from just inside the area wide of Colgan's dive.
Centre-back Matthew Mills, on loan from Manchester City, was allowed to charge forward before firing in a long-range shot that drifted over the bar.
And at the other end a rare threat from the home side came when Eckersley, on loan from Manchester United, burst down the left and curled in a super cross. However, when Richards stretched to connect he headed over the top.
Barnsley stepped up their performance after the break and a dangerous cross from Martin Devaney was heading for Richards until Wayne Brown's clearance cut out the threat.
Eckersley overlapped on the left and broke forward, linking well with Peter Rajczi before firing in a shot that was wide of the target.
The Hungarian then found a gap on his own and broke into the area before hitting an angled shot over.
Colchester, content to sit on their two-goal advantage, waited for scoring chances to appear and might have stretched their advantage when Duguid met a corner and hit a goalbound shot but it was blocked by Eckersley
The visitors were denied by the woodwork when Johnnie Jackson slid a neat pass for Cureton but his dipping shot hit the bar and he then repeated the feat with a 71st minute header.
Seven minutes from the end substitute Hogan Ephraim added the third when he dodged a series of challenges and drilled in a shot from 15 yards.