The winger broke in the Saints U21 team at the age of 15.
Robinson said: "It's been a long time coming, waiting for this moment my whole life.
"Playing up with the older boys has made me more mature, so I think it's definitely benefitted me. I've progressed a lot in the last two years, so it's been good."
Academy director Andy Goldie also told the club's website: "We're delighted that Jay has become the latest high-potential talent to commit his future to the club, matching our belief in him with commitment to his long-term development.
"He made good progress last season, playing centrally and wide with the Under-18s and Under-21s, and has already impressed Russell and the First Team staff during this pre-season.
"It’s important for him to continue to build on this momentum throughout the season, maximising the opportunities he has here, and we look forward to seeing his continued progress."