The deal will see shareholders in the bank receive 340p per share and state-backed Lloyds selling its 50 per cent share.
TSB chief executive Paul Pester will continue in his current role.
The announcement comes a week after plans by Barcelona-based Sabadell were disclosed, causing a surge in the share price.
It means investors who bought the stock at the offer price of 260p when TSB floated nine months ago will receive a 31 per cent premium.
Sabadell said it was attracted by a UK banking market with “a well-defined and stable regulatory framework, consistent profitability and good future growth prospects”.
It said it was picking up a “straight-forward retail and small business bank” with a 6% share of UK branches, which attracted 8.4 per cent of new and switching personal bank accounts in 2014.
The Spanish bank said that under its ownership, “TSB will be able to further enhance its growth strategy and efficiency, benefiting from Sabadell’s resources, experience in SME lending and experience gained in the Spanish banking market”.
Sabadell said it saw opportunities to continue growth in the current account market, and accelerate growth in lending and expansion in the small business sector.
It said potential savings of £160 million a year could be made through a switch over of IT systems from those currently provided by Lloyds to Sabadell’s own platform. Lloyds will provide £450 million to support the transition.