The 20-year-old has been on loan with the Hatters this season, playing in the Johnstone's Paint Trophy win at Wembley, and impressed Norwich boss Bryan Gunn enough to earn a new two-year deal.
Spillane told Norwich's official website: "I'm delighted to sign my contract and stay at Norwich - I came through the youth system and it's great to carry on here.
"Luton were relegated on Monday so that's at the back of my mind but I'm very happy to be coming back to Norwich next season.
"Being with Luton has done me the world of good. I've been playing every week and loved my time with them."