Russell, 30, joins Bath on a reported £950,000 ($1.1 million) following the 2023 Rugby World Cup next November ending a five-year stay at Racing.
The French club said in a statement both parties had failed to come to agreement on the financial details to renew his contract which comes to an end this season.
The Times newspaper reported Russell, who had also attracted interest from Japan, was unwilling to bend to Racing's demands to take a £200,000 pay cut.
Russell has scored 272 points in 66 Top 14 appearances and was pivotal in guiding Racing to the 2020 European Champions Cup final where they lost to Exeter Chiefs.
A lot will be expected of him at Bath as they are a long way off their glory days and this season are just one point better off than bottom club Bristol Bears.
"Finn is a world-class player who will significantly add to our squad on and off the field," said van Graan in a Bath club statement.
"To be able to attract players of Finn's quality to Bath Rugby is a credit to how far we have come as a club and a group since 11 July (when van Graan took over)."
The 65-times capped Russell, who has had an up and down relationship with Scotland coach Gregor Townsend, said it was partly the good things he had heard about South African van Graan and his ambitious plans for the club that had persuaded him to join.
"To be able to join such a historic, well-supported club is really exciting," said Russell.
"I know players at Bath who speak highly of the environment, and the direction the club is going in is something I want to be a part of.
"I've enjoyed my time at Racing 92 and I've learned a lot.
"Now is the right time for a new challenge, in a new league, with the aim of moving Bath back to the top end of the Premiership and competing for domestic and European titles."
