The 20-year-old has agreed a season-long loan with Daniel Farke's side in search of regular first-team minutes.
He signed for Chelsea from Rangers in 2017, and emerged through the academy ranks at Cobham and burst onto the scene last year after a dazzling display in his side's 2-0 victory over Liverpool in the FA Cup.
After warming the bench for a large chunk of Thomas Tuchel's reign at Stamford Bridge, the young Scot will stay in the Premier League and develop further by getting first-team minutes under his belt.
After the confirmation of the loan switch, a few Chelsea stars wished Gilmour the best for his next chapter on Instagram, including Callum Hudson-Odoi, Ben Chilwell and Tammy Abraham.
Gilmour managed 14 outings across all competitions for the London side last season, with the likes of N'Golo Kanté, Mateo Kovačić and Jorginho often being given the nod in the middle of the park.
After putting in a series of compelling displays for the academy, Gilmour was promoted to the first-team by former Chelsea boss Frank Lampard.
However, he suffered a nasty knee injury towards the end of the 2019/20 campaign, which saw his spend four months on the sidelines. He made a swift return from injury in November last year, featuring several times in the Champions League group-stage.
A gifted midfielder as any amongst his age, Gilmour will flourish at Norwich and could challenge for a first-team spot when he returns from his loan spell next summer.