The Villans took the lead in the seventh minute through Scott Hogan, who reacted quickest when the League Two side's goalkeeper Sam Walker spilled a through-ball into his path, placing his strike into the bottom corner.
Colchester, however, grew into the game after going two down and started putting their opponents under the cosh, causing them all kinds of problems in the final third.
John McGreal's men pulled a deserved goal back in the 39th minute when Kent struck a screamer into the bottom corner from 25 yards out, beating Steer at full stretch.
Colchester went in search of an equaliser after the break, but while committing men forward they almost fell further behind, Adomah's shot from close range needing a line clearance from Kent to stay out.
The Os' best chance to equalise came eight minutes from time when Szmodics gained possession on the edge of the box and let fly, forcing an acrobatic save from Steer, who turned the ball behind with a strong hand.
Szmodics's strike proved to be the last meaningful opportunity of the match as the Villans ran the clock down and booked their place in round two.