The talented 21-year-old has signed a four-year contract with the English Championship side in a deal worth £4.3m.
Diallo was snapped up by Metz from Academie Generation Foot in 2023 and last season contributed seven goals and four assists as the team secured promotion to the French top flight.
He becomes head coach Liam Manning's ninth signing of the transfer window.
After signing for the Canaries, a delighted Diallo expressed his excitement with the move.
He said on BBC: "It has been my dream to play in England. I really always wanted Norwich City.
"I am excited about the project and the vision for this club and I can't wait to come to England and meet my new team-mates and the supporters."
Norwich sporting director Ben Knapper added: "Papa is a huge talent, and we're delighted to have won the race for his signature.
"He's a player with enormous potential and he'll add pace, and a real directness, to our attacking options."
Senegal will have to wait until the end of November to discover whether they can reclaim the Africa Cup of Nations (AFCON) title after the Court of Arbitration for Sport (CAS) confirmed that its ruling on their appeal would take several more weeks.
Senegal head coach Patrick Vieira has welcomed the pressure from supporters, insisting their desire to see the national team compete at the highest level matches his own ambitions.
Senegal international Krepin Diatta has secured his next move after joining Championship side Watford on a free transfer following his departure from French club AS Monaco.
In Tuesday's AFCON 2027 qualifiers, Egypt put five goals past South Sudan, Guinea-Bissau shocked Nigeria 3-0, and Ivory Coast beat Somalia.
