The deal is the latest in a string of big overseas acquisitions by Indian companies in recent years, including Tata Motors Ltd's $2.3 billion purchase of Jaguar Land Rover and mobile operator Bharti Airtel Ltd's $9 billion takeover of the African operations of Kuwait's Zain.
"It is very important for us to expand our horizons. Especially in the long run, the U.S. market is going to look up ... ," Apollo Chairman Onkar Kanwar said.
Indian car sales fell 7 percent in the financial year that ended in March, the first annual fall in a decade, while sales in Europe, Apollo's second-largest market, are at a 20-year low.
But auto sales are one of the bright spots for the U.S. economy. Sales rose more than expected in May as construction workers and oil drillers bought more pickup trucks, and they are expected to remain strong for the rest of the year.
Kanwar has expanded Apollo, which had revenue of $2.5 billion in 2012, after taking control of the company in 2002, following a prolonged public spat with his father, who founded the company in 1976.
His previous acquisitions include South Africa-based Dunlop Tyres International Ltd in 2006 and Dutch tire-maker Vredestein Tires in 2009.
Reuters and others reported in October that Apollo and Cooper were in talks for Apollo to take a stake in the Findlay, Ohio-based company.
Apollo said it would raise $2.5 billion in new debt to fund the deal, of which $2.1 billion would be through the issue of dollar bonds with a tenure of seven to eight years.
The deal values Cooper at 4.4 times the its EBITDA (earnings before interest, tax, depreciation and amortisation), which Apollo said was within the range of 3.5 to 6 times multiples seen in recent transactions in the sector.
Apollo said it would launch some Cooper brands in the Indian market, but did not give a specific timeframe.
Apollo shares closed up 2.7 percent at 91.95 rupees in Mumbai trading ahead of the announcement.
Shares of Cooper's larger rival Goodyear Tire & Rubber were up 4.5 percent at $15.27.
Morgan Stanley, Deutsche Bank Securities and investment firm Greater Pacific Capital advised Apollo on the deal. BofA Merrill Lynch was financial adviser and Jones Day was legal adviser to Cooper.