Until its discovery, the cavern's existence was just a rumour, with it believed to lie below one of three gardens where the excavations have taken place.
David Spurling has led digs at the site for six years and said being able to invite the community to get involved has kept the group "motivated".
Mr Spurling said: "We always thought there would be a stream to provide water to the cloisters but it's the first time we've found proof.
"In doing these digs we've come across a lot of fantastic finds and the great thing is as a community group we can get children involved - the feedback we get is fantastic."
It is believed the tunnel would have been used to supply water to the refectory, kitchens and infirmary of the abbey, as well as the latrines associated with the monks' dormitory.
As well as the the discovery of the tunnel, the dig also returned to a garden where uncovered remains of a two-storey building were found in 2020.
Medieval window glass was found in the remains of that building and was being evaluated by Historic England.
Excavation work in all three gardens also found large numbers of Roman pottery, medieval tiles and other building materials as well as oyster shells and bones.
Archaeologist Professor Martin Biddle, who has been tracking the Hyde900 digs, said: "The discovery of a sophisticated example of a vaulted channel, probably late medieval, is extraordinary."