Cloud computing is the increasingly popular practise of using remote internet servers to store, manage and process data.
Apigee specializes in managing so-called application programming interfaces, or APIs, the channels through which digital services connect when a company logs a purchase for a customer or places an order with a supplier.
Google will pay Apigee shareholders $17.40 for each share, a 6.5 percent premium to the stock's Wednesday close.
Apigee's shares were slightly above the offer price at $17.43 on Nasdaq in afternoon trading on Thursday.
The company, whose customers include AT&T, Burberry Group Plc, Vodafone Group Plc and the World Bank, went public in April last year at $17 per share.
Greene, a former VMware CEO, has pushed to raise Google's profile in corporate computing since she joined last year.
During her tenure, Google has streamlined engineering efforts and appointed new leadership for its cloud efforts, improving traction with clients, Google Chief Executive Officer Sundar Pichai said during the company's latest earnings call.
Greene predicted that the Apigee acquisition would redouble Google's momentum.
"Our customer lists are extremely complimentary," she said. "There's some overlap and some areas where we are going to be able to help each other once the deal closes."
The Apigee deal comes a day after Google and online storage company Box Inc said they would partner to enable Box's corporate customers to integrate Google's suite of word processing, spreadsheets and other productivity tools, known as Google Docs.
Google, Amazon.com, Microsoft, IBM and others are vying for a share of the fast-growing corporate cloud computing business.
Apigee, with high-profile clients in a strategically important area, will help Google close in on the competition, said analyst Patrick Moorhead of Moor Insights & Strategy.
Google has fallen behind both Microsoft Azure and Amazon Web Services in enterprise cloud computing, and this move is intended to strengthen that position, he wrote in an email.