The highly-rated 18-year-old has joined Jackie McNamara's squad after penning a three-year deal.
McNamara said: "Charlie is a very gifted footballer with great awareness and excellent delivery from set-pieces. His style of football will settle in perfectly with the rest of the young talent we are currently producing at the club.
"He is a sensible, level-headed young man who wants to play for Dundee United and I am delighted that we have secured his services for the next three years."
Rangers had rejected United's initial offer of compensation for the teen but Telfer - handed his first-team debut by Ibrox boss Ally McCoist in a League One clash with Stenhousemuir in April - will now set his sights on joining the Arabs' growing list of impressive youngsters.
He said: "I have watched how the younger players at United are allowed to develop in the first team and that has been a big reason in my decision to join the club.
"The opportunity to play in the Premiership next season was also something I could not turn down. When I met the manager and saw the facilities at St Andrews, I know that I can further my football development at United."