Microsoft Corp. said its earnings continued to grow last quarter as its cloud-services business stayed strong.

On Tuesday, the Redmond, Wash., software giant said its sales in the quarter ended in December hit $51.7 billion, up 20% from a year earlier. Its net income rose 21% to $18.8 billion.

The results beat predictions from analysts, who...

Microsoft Corp. said its earnings continued to grow last quarter as its cloud-services business stayed strong.

On Tuesday, the Redmond, Wash., software giant said its sales in the quarter ended in December hit $51.7 billion, up 20% from a year earlier. Its net income rose 21% to $18.8 billion.

The results beat predictions from analysts, who were expecting $50.7 billion in revenue and $17.5 billion in net income, according to FactSet.

Microsoft’s overall cloud revenue increased 32% from the year-earlier quarter to $22.1 billion. Its cloud-infrastructure service, called Azure, grew by 46%, down slightly from the prior quarter’s 48% growth

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Microsoft shares initially fell in after-hours trading, but then rebounded to a gain of more than 1% after the company issued its outlook for the current quarter.

For the period, Microsoft expects sales of $48.5 billion to $49.3 billion, compared with analyst expectations of $48.1 billion, according to FactSet.

Over the past two years, Microsoft has been one of the biggest beneficiaries of the shift of remote working that followed the spread of Covid-19. Companies, governments and schools across the world started using more cloud-based technologies, and Microsoft has been selling the tools used for the shift. While an increasing number of organizations have been returning to doing more work in offices, the Omicron variant of the virus may be delaying the return to normalcy.

The cloud-services business is one of the fastest-growing in tech and could expand from $385 billion in 2021 to $809 billion by 2025, according to research firm International Data Corp. Microsoft is the second-largest player in the industry with a nearly 20% share of the global cloud-computing market after Amazon.com Inc., which dominates the sector with a more than 40% share, according to Gartner Inc.

Demand for personal computers, many of them using Microsoft’s Windows operating system, has climbed in recent years, as has Microsoft’s videogame business. Demand for laptop and desktop PCs is expected to continue to grow.

Windows licensing revenue from PC makers increased 25% in the latest quarter, while the company’s Surface computer sales rose 8%.

Microsoft has been making big moves to boost its gaming business. Last week, it announced a $75 billion offer to acquire gaming company Activision Blizzard Inc. If the deal goes through, it will be Microsoft’s biggest acquisition by far.

“With our planned acquisition of Activision Blizzard announced last week, we’re investing to make it easier for people to play great games wherever, whenever and however they want. And also shape what comes next for gaming as platforms like the metaverse develop,” Microsoft Chief Executive Satya Nadella told analysts in a conference call after earnings were announced on Tuesday.

Microsoft’s gaming revenue in the quarter grew 8% from a year earlier, with its hardware sales advancing 4% on demand for the latest Xbox consoles. Revenue from gaming content and services climbed 10%.

The Activision deal, if completed, would ratchet up the size of Microsoft’s videogame empire, making it the third-largest gaming company in the world by sales and adding popular game franchises including Call of Duty and Candy Crush to Microsoft’s Xbox console business.

The offer for Activision also highlighted Mr. Nadella’s mission to reshape the company by leveraging its cloud infrastructure to grow in new markets. The acquisition would bolster its efforts to entice consumers to its already fast-growing cloud-gaming service. Microsoft’s subscription-gaming service, Game Pass, hit 25 million subscribers recently, up around 39% from a year ago.

With more gamers playing on smartphones than on pricey game consoles and computers, Microsoft is leading the race to develop services for streaming high-end games to all kinds of devices the same way movies and TV shows are streamed.

Microsoft shares have done well since the start of the pandemic. They are up around 75% over the past two years, and for a short period last year Microsoft ranked as the most valuable company in the world.

This year its shares have slipped as part of a wider market selloff and are down more than 5% since the Activision acquisition was announced last week.

Despite the strong results for Microsoft on Tuesday, some investors and analysts said they were looking for signs that Microsoft’s growth may have peaked.

The company’s shares initially slipped more than 4% in after-hours trading after the announcement of the results. Even though the overall earnings numbers were better than anticipated, some investors had expected more of the quarterly growth to come from the company’s cloud services, said Stifel Financial Corp. analyst Brad Reback.

“The fact that maybe they’re reporting decelerating revenues for their fastest-growing segment is really raising concerns among some of the investors that the pace of growth in that division has really peaked,” said David Wagner, a portfolio manager at investment firm Aptus Capital Advisors, which owns shares in Microsoft.

Write to Aaron Tilley at aaron.tilley@wsj.com