The Suns did not disclose contract terms.
The 6-foot-8 Jackson was selected No. 4 overall last month out of Kansas, where he averaged 16.3 points, 7.4 rebounds, 1.7 steals and 1.1 blocked shots while shooting 51.3 percent from the field in his one and only season with the Jayhawks.
Jackson, the 2017 Big 12 Freshman of the Year, was only the fourth freshman to average at least 15 points, five rebounds and three assists and shoot better than 48 percent from the floor, joining Ben Simmons, James Harden and Dwyane Wade.
Jackson will join the Suns' NBA Summer League team in Las Vegas for his first action as a professional.
The release of the 14-year veteran saves the Suns $3.5 million and pushes Phoenix's cap space to nearly $12 million.
Barbosa averaged 6.3 points, 1.6 rebounds and 1.2 assists in 67 games last season with the Suns. The 34-year-old, who has played in three stints with Phoenix, ranks 13th in franchise history with 6,443 points and first with 4,825 scored when coming off the bench.
Phoenix is collecting a young group of talent that includes Jackson, Devin Booker, T.J. Warren, Marquese Chriss and Dragan Bender.