Apple has long performed like clockwork, growing steadily and producing an ever-growing stream of profit. Not anymore.

On Tuesday, the Silicon Valley behemoth said that its net income had fallen 13 percent and that its revenue rose 1 percent in the latest quarter, with iPhone sales continuing to decline and gains in the company’s services and wearables business failing to make up the difference.

The results showed persistent signs of weakness for one of the world’s financial standouts. Apple built its enormous business on the iPhone, but sales of the device have slipped for three straight quarters in a saturated market for smartphones.

Yet the results also suggested that the company could be starting to halt declines in those sales and other key areas, including revenue from the Chinese market. Over the previous two quarters, Apple’s profits and revenue had fallen over all.

“Obviously on the iPhone, we’ve gone through a period where we’ve seen some revenue declines,” Luca Maestri, Apple’s finance chief, said in an interview. “But we are very excited about our product road maps, and we’re very optimistic about the future.”

Apple said net income had dropped to $10.04 billion for its fiscal third quarter, from $11.5 billion a year earlier, with profit of $2.18 a share exceeding Wall Street estimates. Revenue rose to $53.8 billion from $53.3 billion a year earlier. Mr. Maestri said profits had fallen while sales had risen because of narrower margins that he attributed to foreign-exchange rates.

Apple shares rose more than 4 percent in after-hours trading. “They essentially beat fairly low expectations,” said Angelo Zino, an analyst at CFRA Research.

Mr. Maestri said Apple had slowed the bleeding in its iPhone business by offering financing, cutting prices in some countries and starting a trade-in program for owners of older models after finding that people were keeping their iPhones longer. In the latest quarter, revenue from iPhone sales fell nearly 12 percent, to $25.97 billion, from a year earlier. In the company’s previous quarter, iPhone sales fell 17 percent.

For the first time since 2013, iPhone sales did not account for at least half of Apple’s revenue, said Yoram Wurmser, an analyst at the market-research firm eMarketer.

Consumers are finding fewer reasons to upgrade their iPhones, analysts said, with newer models offering only incremental improvements. The trend could continue this year, when Apple is likely to unveil a new slate of iPhones. The latest models, expected in September, are unlikely to work with the new fifth-generation, or 5G, wireless technology that offers far faster download speeds than current service. Apple is expected to have 5G iPhones for 2020, analysts said.

The other shrinking part of Apple’s business has been China. Sales in the region that includes China fell nearly 25 percent over the previous two quarters, sparking a sell-off of Apple shares in January. Apple blamed the drop in part on economic weakness in China.

In the latest quarter, Apple’s sales in the region fell 4.1 percent, while revenue specifically in mainland China grew. Timothy D. Cook, Apple’s chief executive, said on an earnings call that business had been lifted in part by Apple’s move to cut iPhone prices there and an economic stimulus program from the government.

“There were a lot of questions and uncertainty around China, so the fact that the year-over-year growth rate has improved is likely a relief for investors,” said Toni Sacconaghi, an analyst at Bernstein.

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The Chinese market, Apple’s third largest for sales, has emerged as one of the company’s greatest vulnerabilities. This month, Chinese officials disclosed that the country’s growth had fallen to its slowest pace in three decades. Apple also assembles most of its products in China, which has drawn the ire of President Trump, who has publicly pressured Apple to build more products in the United States.

Since 2013, Apple has made its top-of-the-line Mac Pro desktop computer in Texas. Doing so led to headaches that delayed the computer’s launch. In June, The Wall Street Journal and The New York Times reported that Apple would shift assembly of its new Mac Pro to China.

Last week, Apple filed requests with the United States trade representative’s office asking that components used in the Mac Pro, like power cables and circuit boards, be excluded from tariffs. Apple said in the requests that it could not find the products outside China.

Mr. Trump responded on Twitter that Apple “will not be given Tariff waiver, or relief, for Mac Pro parts that are made in China” and that the company should “Make them in the USA, no Tariffs!”

On Tuesday, Mr. Cook said Apple still wanted to make the Mac Pro in the United States and suggested that exclusions from the tariffs could help make that happen.

“We’ve been making the Mac Pro in the U.S.,” he said. “We want to continue to do that.”

Mr. Trump has placed tariffs on $250 billion worth of Chinese goods, but so far, Apple products have largely escaped the tariffs’ effect. Mr. Cook has encouraged officials in the United States and China to resolve the trade dispute, but while the countries resumed trade talks this week, hopes for a transformative deal are dwindling.

Apple faces other issues in Washington, including antitrust concerns. Last week, the Department of Justice said it was opening an antitrust review of the Big Tech companies. Apple has come under particular scrutiny for how it wields power in its App Store, where it distributes games, ride-hailing programs and more.

As Apple’s iPhone sales fall, the company has sought to make up the gap in revenue with an expanding business selling apps and services to its existing customers. Apple now offers subscriptions for news, music and TV services and is preparing to start a gaming service soon. Its services revenue rose more than 12 percent to $11.5 billion.

Apple also showed strength in its wearables business, which includes Airpods and the Apple Watch. That business grew nearly 64 percent to $5.5 billion, surpassing the iPad in sales.