The former Watford and Fulham boss has signed a three-year contract with the Blades, succeeding Chris Wilder, who left the club in March after nearly five years in charge.
Blades' Under-23 coach Paul Heckingbottom oversaw the club's final 10 matches of the season but was unable to prevent United from making the drop.
Jokanovic won promotion from the second-tier Championship at both Watford and Fulham and the Blades hierarchy will be hoping he can repeat the feat at Bramall Lane.
"I'm excited to work in English football again and I'm looking forward to getting started with the players and staff, as well as meeting our passionate, loyal supporters as we prepare for the challenge in the Championship," said the Serbian.
"I want to assure our amazing fans that we're fully committed to helping the team achieve its goals and make you feel proud."
The club's Saudi owner, Prince Abdullah bin Mosaad bin Abdulaziz al Saud, said: "Once we held discussions with Slavisa, it was also abundantly clear that his ambitions were aligned with those of the club.
It goes without saying, we are thrilled to have someone in charge with a CV which boasts a proven track record of success in England and abroad."