With extra-time looming if the scores were level after 90 minutes, Wivenhoe found a winner just four minutes from time when Myles Pressley's deflected strike crept home.
There was no time left for 'Side to equalise as Wivenhoe advanced to the Second Round of the competition where they will face White Ensign or Holland in the next round.
Michael Walther's Sky Blues are next in action on Saturday in the league when they host St Margaretsbury. Kick-off at Cricklefields is at 3pm.