Microsoft Corp. extended its streak of wins in the latest quarter and signaled further strength in the months ahead as the software giant moves into an era where its venerable Windows franchise plays a supporting role to its burgeoning cloud-computing operations.

The cloud business, called Azure, jumped 93% in the fiscal third quarter, Microsoft reported Thursday. The business has never grown slower than 90% since the company began reporting the metric in October 2015. The other big piece of the company’s cloud operations,...