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."
Crystal Palace manager Pierre Sage has revealed that Ismaila Sarr was keen on joining Liverpool but says the club intend to keep the Senegal international at Selhurst Park beyond the January transfer window.
Pape Matar Sarr has completed a season-long loan move to Juventus from Tottenham, with the deal set to become permanent if certain performance-related conditions are met.
Senegal winger Ibrahim Mbaye has completed a 47 million pound move from Paris Saint-Germain to Aston Villa.
Patrick Vieira has been appointed Senegal head coach as a replacement for the sacked Pape Thiaw following a difficult 2026 FIFA World Cup finals.
