"Now it's time to start working hard, getting back to match fitness as soon as possible and then to let my rugby do all the talking."
Saracens director of rugby Brendan Venter said: "Gavin is a special talent and we are delighted he has chosen to launch the next phase of his career with us.
"We believe he will enjoy our environment, on and off the field - that will be great news for our supporters and for Welsh rugby."
Henson, who signed for Sarries on Wednesday evening, will return to action after completing his commitments on Strictly, which could potentially take him through to Christmas.
However, he is considered one of the outsiders to win the competition and could be voted off at any time, freeing him up to return to rugby.
Henson gets hard time from Strictly judges
In a statement Saracens thanked the Ospreys, and managing director Mike Cuddy in particular, for their "constructive approach in agreeing to release Henson from his contract in the wider interests of the player and of the game".
Henson's contract with the Ospreys had been due to run until May and Cuddy said the club had "reluctantly" agreed to release Henson.
"It is necessary that we make the decision that we have in order to allow everyone to move forward," he added.
Henson, who can play at fly-half, full-back and inside centre, has not played since injuring his ankle in March 2009 and has been on unpaid leave from the Ospreys since August 2009.
Saracens, last season's losing Premiership finalists, want the double Grand Slam winner to fill in for
injured fly-half Derick Hougaard,
who is set to miss the rest of the season with an Achilles tendon injury.
"I am disappointed and frustrated, both personally for myself and for the Ospreys, as one thing we would all want to see is Gavin Henson back in an Ospreys shirt," added Cuddy.
"All rugby fans will look forward to seeing Gavin back on a rugby pitch, and we sincerely hope that he will be back in the red jersey of Wales before too long.
"With the Rugby World Cup taking place next year, no doubt that will be the focus of Gavin's efforts over the coming months.
"However, as frustrating as it is for everyone involved at the Ospreys, on this occasion we have to concede that he will not play for us during the remaining time on his contract, and as such, take the responsible course of action that is best for all concerned."
Cuddy added that Henson had contributed much to the Ospreys and said the club had hoped it would see him return one day.
Henson's wish to play in London was made plain in September,
when he told BBC Sport
he wanted to start playing again, but with a London club so that he could be near his children.
He has won 31 caps for Wales and also toured New Zealand with the 2005 British and Irish Lions.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.