Driver, who becomes the latest first-team player to leave Tynecastle, will join the Dynamo on loan until his Hearts deal expires in the summer.
A statement on the club website read: "The club wishes Andrew all the best as he continues his career with this fantastic opportunity to play in America."
Hearts did not give details of any fee or of the contract he will sign in Houston.
Oldham-born Driver had seemed certain to move on given the club's need to cut their wage budget.
The former England Under-21 international, who can still play for Scotland under the five-year schooling rule, has made more than 150 appearances for Hearts since coming through the youth ranks.
Hearts placed a £4million price tag on Driver in summer 2009 when Burnley attempted to sign the player for a second successive summer, while Coventry and FC Twente were also linked with moves for the player.
But a series of injuries hampered his progress, although he has been fit without interruption for more than 12 months now, and Hearts will have had to accept a much smaller fee from the Americans.
Hearts have sold David Templeton and Ryan McGowan this season after releasing a significant number of experienced players last summer, although they were able to bring in Liverpool pair Danny Wilson and Michael Ngoo on loan in January.
The income from player sales is very welcome for the board as discussions continue over the future control of the club.
Hearts director Sergejus Fedotovas revealed on Saturday that takeover talks were progressing with a number of interested parties while discussions continue over a possible role for supporters, whose £1million purchase of shares helped the club survive before Christmas.
The Foundation of Hearts is still bidding to acquire a majority share for the fans and will hold talks with an interested Scandinavian consortium, but the group today warned that uncertainty over Vladimir Romanov's business interests must be resolved.
Romanov's UBIG group control the club as well as Ukio Bankas, which went into administration in Lithuania last week.
The group's chairman, Alex Mackie, said in a statement: "We are moving forward positively with our plans, but Ukio Bankas being placed in temporary administration and last night's announcement of its insolvency and the permanent termination of its banking licence have added additional concern to an already challenging and confused situation.
"There is a lack of clarity as to the commercial relationships between Ukio Bankas, UBIG and the football club and for our part (as would be the case for any prospective buyer), due diligence dictates that the potential implications for the club have to be clarified.
"We are therefore - with the support of specialists in this field - examining the financial links between the various entities in order to assess the implications.
"This is not a straightforward task but we take the responsibility of potentially being entrusted with our fellow fans' money very seriously and will not take unnecessary or foolish risks.
"We will only table a bid, therefore, when we have sufficient and satisfactory financial and other relevant information.
"Although we signed a non-disclosure agreement with the current owners some time ago, we have very little of the information we requested, while the last publicly available financial information (the audited accounts for the period to 30 June 2011) is now almost 20 months out of date. Our priority now is to establish transparency and clarity.
"As we move ahead, we will also be meeting with the Scandinavian consortium later this week, and we will be reporting back to supporters again after this meeting has taken place.
"The next few weeks will clearly be critical for Heart of Midlothian and we aim to continue to engage with fellow fans to secure - together - the long term future of this great club."
Meanwhile, Darren Barr could face Inverness in the Clydesdale Bank Premier League on Saturday despite suffering a nasty head injury against Kilmarnock last weekend.
Barr was carried off on a stretcher and was given oxygen as well as staples in a wound after a collision of heads with Wilson.
But manager John McGlynn told Hearts News: "The situation is much better than we anticipated.
"The doctor is very positive about Darren, so much so that he could be in contention for the Inverness game on Saturday.
"Darren will continue to be assessed daily, but absolutely no risks will be taken with him."