The 28-year-old, who joined Carnegie from Rotherham in 2011, has scored 28 tries in 138 games at Headingley.
He is the fifth confirmed signing by Falcons for 2017-18, following the Matavesi brothers, Maxime Mermoz and DTH van der Merwe.
"Dean Richards really sold the vision and the ambition," Burrows said.
"They are playing a style of rugby which really appeals to me, and overall it's just an attractive prospect."
Director of rugby Richards said: "Ryan is an experienced performer who has played to a consistently high standard in the Championship over a number of years now, and really deserves this chance to step up."
Newcastle Falcons will sign back-rower Ryan Burrows from Championship side Yorkshire Carnegie at the end of the current season on undisclosed terms.