In manager Ian Flaherty's absence, George O'Boyle was in charge of team matters and made only one change to the side which knocked Blairgowrie out of the same competition the previous week.
Gavin Beith was on from the start with Dougie Craig dropping to the bench where he was joined by Wayne Morrison, Daryn Smith, Garry McLaughlin and spare goalkeeper 'Chipper' Whyte.
Carnoustie took the lead in 34 minutes. A Conway free kick was touched on to BEITH, who cut inside his marker and fired in a low drive, which Brown got his hands to, but could not prevent it from crossing the line.
The Gowfers doubled their advantage in 41 minutes. Conway made a great run up the left and sent in a deep cross to the back post where COOK slid in and fired the ball into the net off the inside of Brown's right-hand post.
Tomorrow (Saturday), Carnoustie will take on Forfar West End in the final of the GA Engineering Cup at Westview Park, Kirriemuir. Kick-off is at the earlier time of 2.15 p.m.
Carnoustie's reward for beating the Bluebells is a semi-final tie at Laing Park against Montrose Roselea on Tuesday evening with a 7 p.m. kick-off.
Meanwhile, the saga of the relegation issue still rumbles on. With Glenrothes having moved clear of the drop zone after a 3-1 away victory over Bathgate Thistle on Saturday, Carnoustie's slim chances of playing East Super League football again next season now rest with the fortunes of Hill of Beath Hawthorn.
In order to stay up, the Gowfers must win their remaining league fixture against Whitburn and hope that the Haws take nothing from their last three fixtures against Glenrothes, Kelty Hearts and Lochee United. It would then come down to goal difference and the Laing Park men would just have the edge over the Fifers.