The Scotland Under-19 international had been with the Ibrox club for more than 10 years.
Telfer made his senior debut for the Light Blues in April in the 4-0 League One win over Stenhousemuir, playing 22 minutes off the bench.
United boss Jackie McNamara said: "Charlie is a very gifted footballer with great awareness and has 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."
Telfer 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 Scottish Premiership next season was also something I could not turn down.
"When I met the manager and saw the facilities that are available at St Andrews, I know that I can further my football development at Dundee United."