"Based on our observations and media reports, launch day supply was largely sold out within the first 10 to 30 minutes depending on model," Piper Jaffray analyst Gene Munster said in a client note.
Munster, who expects Apple to sell 2.3 million watches in the April-June quarter, expects Apple to ramp production between mid-May and June.
Bank of America/Merrill Lynch expects Apple to ship 4 million smartwatches in the quarter.
The Apple Watch is Chief Executive Tim Cook's first new major product launch and the company's first foray into the personal luxury-goods market.
The company will sell the gadgets in Apple stores and online as well as in trendy fashion boutiques in Paris, London and Tokyo as part of a strategy to position the wearable computer as a must-have accessory.
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Despite mixed reviews, which praised the watch's styling but criticized its less-than-spectacular battery life and slow-loading apps, people flocked to Apple's stores to get a close-up look.
"We think the Apple Watch will be highly disruptive to the traditional fashion watch market," Pacific Crest analysts said.
The analysts said their checks showed that Apple appeared to be ordering components for the watch that would allow it to build well over 20 million watches this year.
"While this represents just over 5 percent of the company's iPhone user base, it would be nearly half of the total (more than) $200 watch market," they said.
The brokerage expects Apple to likely have about 5 million Apple Watches available for delivery by April 24.
Analysts widely expect the Apple Watch to outsell similar watches offered by Samsung Electronics, Sony, and LG Electronics, which have attracted lukewarm interest from consumers.