The 21-year-old, who will command a development fee, becomes the Ibrox side's ninth signing of the summer.
The winger scored four times in the English League Cup last season.
"I am very excited, there are great facilities, the lads are great, the manager is great, the staff are great so I just can't wait to get started," Dodoo told the Rangers website.
"I sense a similar spirit to Leicester last season here, the lads were together and here it is exactly the same, the lads are just really good friends.
"That's got Leicester to where they were because they were a really tight group and I see the same thing here."
Rangers manager Mark Warburton, speaking after Tuesday's 2-0 League Cup win over Annan, said: "Joe is like Jordan Rossiter. He's young with great potential - an exciting talent, technically very strong, a good athlete and a good size.
"He will come in and be keen to make an impression and we're delighted to get him."
Dodoo follows Joey Barton, Matt Crooks, Matt Gilks, Clint Hill, Lee Hodson, Niko Kranjcar, Jordan Rossiter and Josh Windass in moving to Ibrox this summer.
Warburton also revealed that right-back James Tavernier is close to signing a new deal, with discussions continuing with striker Martyn Waghorn.
"To get Wes Foderingham and Rob Kiernan to extend their deals by a year is good news," said Warburton.
"Tav is all agreed too, it's an extra year as well. The paperwork will be done in the next couple of days.
"The discussions with Waghorn go on. I know you guys make a big deal out of it but these discussions go on. But Waggy's attitude has been first class."
Forward Joe Dodoo has signed a four-year contract with Rangers after leaving Leicester City.