The transfer was finalised on Thursday with the 22-year-old agreeing a five-year deal after leaving Championship rivals Huddersfield Town.
"An awful lot of work has gone into this deal and we are all indebted to Mrs Desai and her family for their magnificent support," Rovers managing director Derek Shaw said of club owner Anuradha Desai.
Desai is chairperson of Indian poultry empire Venky's.
"From the very first day I walked in here, the owners have given superb backing for everything we are trying to do on and off the field. That is where the credit is really due."
That feeling has not been shared by the fans of the former Premier League winners, who were relegated last season from the top flight.
Supporter protests at the club's home ground, Ewood Park, against the running of the club and the decisions made by manager Steve Kean are a common sight.
Venky's, who purchased the club for £23 million in November 2010 and removed popular manager Sam Allardyce shortly after, have stood by Kean. Rhodes is their ninth signing since relegation in May as they look to bounce back to the lucrative Premier League.
Rhodes has won two caps for his country and scored 40 goals for Huddersfield in their League One promotion campaign last season.
"Jordan is a top class striker and we are absolutely delighted to bring him in," Kean said.
"His record is there for all to see and I believe he can play a major role in our drive for promotion back to the Premier League."