Wivenhoe of the Thurlow Nunn league started the game well, forcing Ryan Karrer in goal into some early saves in the first period.
'Side came back into the game and either side of half-time forced Town's captain and goalkeeper Aaron Reid into some saves of his own.
With extra-time looming if the scores were level after 90 minutes, Wivenhoe found a winner just four minutes Tues 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.
'Side: Ryan Karrer, Kirk McPherson (Terry Anim), Aaron Hunwicks, Euan Taylor-Reid, Ashley Pibworth, Danny Laurent, Glen Warwick (Garjay Thompson), Harry Matthews, Mark Frostick (Thomas Ngegba), Callum Matthews, Mattheus Santos
Subs Not Used: James Briggs, Jedidiah Brown