Patel, who has captained the club for the last two summers, will retire from all forms of cricket after the 2020 season, and looks set for a career as a coach, having worked as a spin-bowling consultant for England during their T20I series in New Zealand last month.
Rhodes, 24, joined Warwickshire from Yorkshire ahead of the 2018 season, and has become a fixture at the top of the order in the County Championship, opening the batting alongside Dom Sibley.
Paul Farbrace, the club's sport director said: "Despite being only 24, Will has quickly become recognised as a senior member of the dressing room and he has demonstrated excellent leadership qualities that have impressed the cricket management team.
"He can benefit from the wealth of experience that Jeetan offers as a former captain and player who, alongside Ian Bell and Tim Ambrose, knows what it takes to win trophies over a sustained period of time with Warwickshire.
"Jeetan is an outstanding cricketer and we're very grateful for the job that he has done as captain over the last two years. He took on the role at a difficult time, but he has played a critical role in securing our place back in Division One of the County Championship while also managing the transition of bringing many young players into the senior ranks."
Rhodes described his appointment as "a real honour" and credited Patel with aiding his development at Edgbaston. "Jeets has been my first captain here at Warwickshire," he said. "He's played a huge role in helping me settle at the club so quickly and in establishing myself here across all formats.
"Following in the footsteps of a true Warwickshire legend is a very exciting prospect and he will be a great support for me as captain."
Patel said the decision to step back from the captaincy had been a joint decision. "It is no secret that the club is held close to my heart, and renouncing the captain's role, as much as I've enjoyed the challenge of being at the helm over such a challenging period, is a result that both management and myself have come to with the club's future in mind.
"With my official retirement from playing all forms of cricket at the conclusion of the 2020 season, Will gives the club the opportunity to continue to transition smoothly into a young, hardworking and successful team with growth at the forefront of the Bears journey.
"I am wholly committed in my duty to support Will and the team wherever possible throughout my last playing season."
Rhodes will captain both the County Championship and 50-over teams, with Sibley continuing as vice-captain, while the club said that an announcement regarding their T20 captain would be made "in due course".