SAN FRANCISCO — Apple’s biggest cash cow, the iPhone, is gaining weight.

Sales of iPhones, including the new, big-screen iPhone 6 models released last month, helped carry Apple to a record-breaking quarter and offset slowing sales of one of Apple’s other major products, the iPad, the company announced Monday. Apple sold 39 million iPhones in the quarter that ended Sept. 27, a significant bump from the 33.8 million it sold in the same period last year.

Apple appears to be gorging on consumer demand for its smartphone. The company’s $8.5 billion in net profit for its fiscal fourth quarter was 13.3 percent higher than the same quarter a year ago. Revenue over the quarter was $42.1 billion, up from $37.5 billion in the same period last year.

“Demand for the new iPhones has been staggering,” Timothy D. Cook, Apple’s chief executive, said in a financial earnings call with investors. He added: “I’ve never felt so great after a launch before.”

But impolite as it may be to point out in the middle of such exuberant returns, those numbers — while enviable for most companies — do pose some risk for Apple, which gets about 70 percent of its profits from the iPhone, said Toni Sacconaghi, a financial analyst for Sanford C. Bernstein research.