The Sarnians took the lead in a one-sided first half at Les Quennevais and looked set to inflict further misery on the Reds before the break, but a second-half revival from the home side kept the result in doubt until the final whistle.
Guernsey player-coach Andy Whalley said his side had made hard work of progressing, but did so deservedly.
'We had a number of great opportunities in the first half and were disappointed not to go in 2-0 up. We started to get shaky towards the end of the first half and then typically, as these games go, Jersey got in the ascendency.
'They had a penalty corner in the last minute and I had a horrible feeling it was going in. Further on in the competition we need to bury our chances.'
Guernsey have been drawn at home to Blackheath & OEs in the third round, which is a repeat of last season's final.
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