In a decision it said was "mutually agreed" with the supervisory board, Morgan will leave the business software maker on April 30 while Klein will stay on as sole CEO.
Morgan joined SAP in 2004 and ran the company's cloud computing activities - which became a vital growth driver for the firm under the leadership of former CEO Bill McDermott.
"With unprecedented change within the world, it has become clear that now is the right time for the company to transition to a single CEO leading the business," Morgan said in the same statement.
SAP released its first quarter results on Tuesday, which showed that customers began to postpone "a significant amount of new business" last month as the economic fallout from the coronavirus pandemic intensified. Revenues over the first three months of 2020 were up seven percent year-on-year at 6.5 billion euros ($7 billion). Net profit reached 811 million euros, up from a 108 million euro loss over the same period last year.