Cognizant plans to finance the transaction through a combination of cash on hand and debt, and has secured $1 billion of committed financing in support of the acquisition.
The deal is expected to close in the fourth quarter of 2014.
"The transaction is expected to be immediately accretive to Cognizant's non-GAAP EPS, excluding one-time transaction costs and adjustments,' said Karen McLoughlin, Cognizant's CFO, said in a statement.
"That earnings benefit is expected to increase over time as we realize significant revenue synergy potential from the combination of these businesses."
Cognizant shares moved lower in premarket trading following the announcement. (Click here for the latest quote.)
Cognizant's health care business, which accounted for about 26 percent of total revenue in 2013, has declined in the last three quarters.
The company provides services such as claims processing, billing and call center operations to insurers, hospitals and some state-run health care exchanges set up under President Barack Obama's Affordable Care Act, also known as Obamacare.
TriZetto provides information technology services, including care management and the administration of benefits. The company said it reaches 245,000 healthcare providers, representing more than half of the insured population in the United States.
Englewood, Colorado-based TriZetto is the latest U.S. health care IT services provider to be acquired as payers and providers of health care seek new ways to cut costs.
"Health care is undergoing structural shifts due to reform, cost pressure and shifting responsibilities between payers and providers," Cognizant CEO Francisco D'Souza said in a statement. "This creates a significant growth opportunity, which TriZetto will help us capture."
The company in August forecast its slowest full-year sales growth in its 20-year history.
Cognizant, whose rivals include Tata Consultancy Services and Infosys, said it expected revenue synergies of $1.5 billion over the next five years from the deal.
The company said the deal would immediately add to adjusted profit on closing, expected in the quarter ending December.
Apax Partners, which acquired TriZetto in 2008, was exploring a sale of the company, sources told Reuters in August.