The deal will affect thousands of Corus workers in Europe
Tata's bid for the European steelmaker, which was created from the merger of British Steel and Hoogovens, beat that of its Brazilian rival CSN.
Britain's Takeover Panel said Tata had won after offering 608p per share, valuing Corus at £5.75bn ($11.3bn).
Corus employs 47,300 people worldwide, including 24,000 in the UK at plants in Wales and England.
Shares in Corus closed up nearly 7% at 601.5 pence in London as investors welcomed the outcome of the bid battle.
It is a two-way street now - India is seeking foreign investment, but Indian companies are emerging investors in other countries
Kamal Nath
India's Commerce Minister
Q&A;: Corus takeover Indian press reaction Check Corus shares
Yet shares in Tata Steel closed down 11% on Wednesday, as investors worried about the deal's short term financial impact on the Indian firm.
Tata, which is based in Mumbai (Bombay), previously said its takeover would not lead to job losses in the first phase.
The takeover will create the world's fifth-largest steel group.
In the UK, Corus has operations at Port Talbot, Scunthorpe, Newport, Corby, Redcar, York, Deeside, Wolverhampton and Rotherham.
'Big player'
Tata Steel's owner Ratan Tata hailed the takeover as "a moment of great fulfilment for all in India".
"When we first bid for Corus, many thought it was an audacious move," he said at a press conference in India.
"Tata has a global scale now.
"This is the first step in showing that Indian industry can step outside its shores into an international market place as a global player."
India's Commerce and Industry Minister Kamal Nath welcomed the deal and said: "It is a two-way street now. Not only India is seeking foreign investment, but Indian companies are emerging investors in other countries."
Analysts say the deal is a gamble by Tata chairman Ratan Tata
The two-way battle for the firm began in October when Tata tabled a £4.1bn bid for the group and, in December, the Corus board recommended a revised £4.7bn offer from Tata.
But, just hours later the board confirmed it had approved a £4.9bn, offer from Rio de Janeiro-based CSN.
Tata eventually outbid its Brazilian rivals.
Wider Indian group
Last year Corus was the ninth-largest steel producer in the world with 18.2 million tonnes of output.
It banked pre-tax profits of £580m on turnover of £10.14bn.
Tata Steel, part of the Indian conglomerate Tata Group, was last year ranked 56th in the list of steelmakers around the world with output of 5.3 million tonnes.
The Tata Group - which owns Tetley tea and Daewoo trucks - has operations in more than 50 countries.
Tata Group Logo

India's steelmaker Tata Steel, of the Tata Group, buys Anglo-Dutch steel giant Corus Group for £6.7 billion ($12 billion), making it the world's fifth largest steel manufacturer. 70-year-old Tata group Chairman Ratan Tata, from one of India's best-known business families, won the race against Benjamin Steinbruch, 52, a famous Brazilian executive who is the chief and main owner of Companhia Siderurgica Nacional (CSN). Tata paid investors 608 pence a share, whereas the Brazilians final offer in an auction by the U.K.'s Takeover Panel was 603 pence.

The deal triples Tata Steel's capacity to almost 28 million tons a year. Tata: "This is the first step in showing that Indian industry can step outside its shores into an international market place as a global player." 

Corus, which was created from the merger of British Steel and Hoogovens, currently employs 47,300 people worldwide. Last year the company was the ninth-largest steel maker worldwide. 

The takeover may start a round of consolidation in the fragmented steel sector.