Penalty shoot-out earns passage to the next round.
The non-leaguers were reduced to nine men by the time that the final whistle went but had drawn level with ten men still on the pitch on the ninetieth minute.
But their endeavour was not quite enough to progress past a determined U's side who kept their nerve to win the penalty shoot-out.
The first half was one of limited clear cut opportunities, although the U's went close on a number of occasions with shots around the edge of the box, with Szmodics in the thick of it throughout.
It was the Islanders who took the lead on 37 minutes though, as Curran reacted quickest when a free kick dropped to him in the six yard box.
Jack Curtis cracked one against the crossbar before half time, but it was the second half before the U's drew level.
Tosin Olufemi's shot from outside of the box was well saved, but Szmodics drilled it back into the danger area for Sheehan to turn into his own goal.
The U's were further boosted when the visitors were reduced to ten men as Stokes and Lapslie clashed and the Canvey man was sent off for his part in what developed into a fracas between the two sides.
The U's capitalised when Szmodics seized on a loose ball and worked some space to drill home the second.
If the home side thought that'd be enough, Canvey thought otherwise and drew level late on in the game when Poole's free kick clipped two players on the way through to leave Bransgrove bamboozled as it beat him into the bottom corner.
Dumas saw red for a second yellow in injury time, so it was a 9 v 9 shoot out which nearly went all the way.
Lapslie missed for the U's during the drama of penalties, but it was Sheehan's miss in the sudden death stages that ultimately separated the two sides and earned the U's safe passage to the next round.