The boost to Apple's share came primarily at the expense of Google and its Android operating system.
"While remaining the dominant global OS, Android's market share dropped in most European markets and in the U.S., where the decline was the first since September 2013," said Carolina Milanesi, Kantar's chief of research, in a statement Wednesday.
In the U.S., iOS's share climbed from 43.1% in October to 47.4% in November, Kantar reported. The firm regularly surveys consumers in several countries to gauge smartphone ownership, and its figures represent the most recent three-month period. November's data, for instance, is actually for the September-November stretch.
Android's U.S. share fell from 50.4% to 48.4% -- putting Google's operating system just one point ahead of iOS -- while Microsoft's Windows Phone share slumped to 3% from 4.6% the month prior.
The trend in the U.S. was mimicked elsewhere.
In the U.K., iOS gained 3 percentage points as Android lost 2.6 points; in China, iOS increased its share by 2.4 points, while Android's fell by 2.5 points; and in Italy, iOS grew share by 3.5 points as Android slipped by 2.2 points.
Year-over-year comparisons also showed that Apple gained ground in most markets. iOS's share climbed by 4.1 percentage points in the U.S. in the last 12 months, 12.2 points in the U.K., 1.1 points in China, and 6 points in Italy, for example.
The iPhone 6, the smaller of the two newest models, accounted for 19% of all smartphone sales in the U.S., said Milanesi, making it the most-purchased device. In an interview, she said that the iPhone 6 was beating the larger iPhone 6 Plus by at least 3:1 in U.S. sales.
But Apple's gains were not exclusively due to the new iPhones. "Yes, the changes in design [of the iPhone 6 and 6 Plus] have been well received," said Milanesi. "But market share doesn't always come from the top-selling phone. The iPhone 5S is still well."
More consumers who already had an iPhone bought another Apple smartphone than Android owners stuck with that ecosystem, Kantar said, and intent-to-purchase numbers were also higher for Apple than for Samsung in the U.S.
Keeping existing customers in the fold, and grabbing new customers at the same time, is "absolutely crucial" for any smartphone maker, said Milanesi, who added that the more devices a company can sell to any one consumer, the more likely that customer will stick around. "That just makes it that much harder to move away [from an operating system] and that much more likely they'll still around," said Milanesi.
Apple will report iPhone sales for the quarter ending Dec. 31 at its next earnings call, scheduled for Jan. 27 after the market closes. Financial analysts expect Apple will announce record iPhone sales for the period.