Evans, 28, a former presenter of Channel 4's Big Breakfast and currently host of Don't Forget Your Toothbrush, will join on 24 April.
Wright's shock resignation last month prompted talk of a crisis at the station, but this appointment will help to ease the pressure on Matthew Bannister, controller of Radio 1, who has faced strong criticism for declining ratings.
Describing Evans as a "tremendous talent", Mr Bannister said: "It is no secret that I have been intent on luring Chris back to radio since I arrived at Radio 1. This has been the mother of negotiations and it is a fantastic coup for the station. I believe this will mark the beginning of a new chapter in Radio 1's history."
Mr Bannister personally concluded the eight-month deal early on Sunday. It is thought the contract will make Evans Radio 1's highest-paid DJ - figures of between £325,000 and £400,000 have been mentioned. A Radio 1 spokeswoman would only say: "It's a substantial deal because he is a big talent."
Evans said: "There is currently no job in television that matches the challenge presented by Radio 1."
The Breakfast Show will be independently produced by Evans's company, Ginger Productions - set up to capitalise on his success with the Big Breakfast, which he left last September. He started his radio career as a producer at the BBC's Greater London Radio, before moving into presenting. He hosted a Sunday show on Radio 1 until three years ago.Reuse content