The deal gives Apollo access to markets in the U.S. and China while Cooper gets a premium price per share and some assurances that its domestic operations won't be gutted.
Cooper CEO Roy Armes said the two companies are a good strategic fit because each is in different markets.
"The lack of overlap is where the opportunities really are," Armes said in an interview.
The deal will give Cooper's customers access to new products such as off-road and farm tires, which could eventually be made at one of the company's North American plants, he said. "It won't make sense to import everything from India," Armes said.
Apollo has made a "pretty strong commitment" not to shut down plants in Findlay in northwest Ohio, Tupelo, Miss.; and Texarkana, Ark., Armes said.
It also plans to honor the terms of labor contracts currently in place and generally maintain pay and benefit levels for non-union employees.
Cooper executives are expected to continue leading the company from its headquarters in Findlay, but details of how that will work haven't been finalized.
Under the terms of the deal, Cooper shareholders will receive $35 per share in cash. The price represents a 42 percent premium over Cooper's Tuesday closing stock price and is higher than the share price over the last decade.
The companies had a combined 2012 sales of $6.6 billion.
Based on the company's 63.3 million outstanding shares, the deal is worth about $2.22 billion. The companies valued the sale at about $2.5 billion.
Cooper shares jumped $10.10, or 41 percent, to close at $34.66 Wednesday.
Standard and Poor's equity analyst Efraim Levy said that the biggest question was how Apollo will finance the purchase of the larger company. "It seems to be that they're biting off a lot," he said.
He said that Apollo is paying such a high premium per share that it would have been hard for Cooper to reject the offer.
Apollo Chairman Onkar Kanwar said the combined company will serve both large, established markets, such as the U.S. and Europe, as well as fast-growing markets such as India, China, Africa and Latin America.
With a history dating back to 1914, Cooper currently employs nearly 13,000 people around the world and has manufacturing plants on three continents. Its brands include Cooper, Mastercraft, Dean, Starfire, Roadmaster and others.
Its 2012 its revenue totaled $4.2 billion. Last month, the company said its first-quarter profit more than doubled to $56.1 million, as lower raw material and manufacturing costs more than offset a double-digit drop in sales stemming from lower global demand and tough economic conditions.
Apollo, founded in 1972, produces premium and mid-tier tires in a variety of brands including Apollo and Vredestein.
The sale, which remains subject to Cooper shareholder and regulatory approvals, is expected to close in the second half of this year. When that happens, Cooper will become a privately held company and cease trading on the New York Stock Exchange.