London-born Williams, 24, has penned a two-year deal with the Tigers, with the club having an option of a further year.
Williams, who will officially join on July 1 when his contract with the Grecians expires, has been the subject of prior interest from City, who saw a bid turned down by the Devon outfit last summer.
The East Yorkshire club have now finally landed the long-time target, who had also been linked with several other clubs.
Williams was Exeter's player of the season in 2019/20 after helping the Grecians reach the League Two play-off final. He was also named in the division's team of the year.
On signing former Wycombe winger Williams, who had spells at the start of his career with Crystal Palace and Watford, Tigers head coach Grant McCann said: “We’re absolutely delighted to eventually get Randell. We offered a big fee for him this time last year and the boy was very keen to come to us.
“Unfortunately, Exeter weren’t prepared to let him go, so to get him now on a free transfer is quite a coup for us.
“Randell’s got electric pace, loves getting at defenders in one-v-one scenarios, can go both ways and will provide serious competition in the forward areas of the pitch.
“Most of all, he’s a very humble, down-to-earth lad who again fits our model of young and hungry players who want to learn and improve. We believe in this environment, he will do just that.”
Williams' addition follows on from the captures of George Moncur and Andy Cannon.