The rare apology follows Apple's launch of its own mapping service earlier this month, when it began selling the iPhone 5 and rolled out iOS 6, the highly anticipated update to its mobile software platform.
Users complained that the new Maps service - based on Dutch navigation equipment and digital map maker TomTom NV's's data - contained geographical errors and gaps in information, and that it lacked features that made Google Maps so popular, including public transit directions, comprehensive traffic data or street view pictures.
"We are extremely sorry for the frustration this has caused our customers and we are doing everything we can to make Maps better," Cook said in a letter to customers released on its website, adding that the company "fell short" of its commitment to deliver "the best experience possible to our customers."
He then suggested that customers download rival mapping services available in Apple's App store, while the company improves the product.
"While we're improving Maps, you can try alternatives by downloading map apps from the App Store like Bing, MapQuest and Waze, or use Google or Nokia maps by going to their websites and creating an icon on your home screen to their web app," he said in the letter.
Apple is typically loathe to tout rival services and the contrite apology by Cook is an indication of how Apple is changing under the chief executive who took over last year from co-founder Steve Jobs just before his death.
The last time Apple faced such widespread criticism was in 2010, when users complained of signal reception issues on the then-new iPhone 4 model.
A defiant Jobs at the time rejected any suggestion the iPhone 4's design was flawed but offered consumers free phone cases at a rare, 90-minute press conference called to address those complaints.
While Apple fixed the issue, Jobs had apologized to users only after he was specifically asked if he was sorry.
Cook's suggestion that Apple users download other map apps, particularly Google Maps, represents an about-turn for Apple, which had introduced its service as a direct challenge to the popular service offered by ally-turned-rival Google.
Apple Maps replaced the Google Maps app, which is no longer available on iOS 6. Users now have to access Google Maps through the browser.
Google and Apple were close partners with the original iPhone in 2007 and its inclusion of YouTube and Google Maps. But ties between the two companies have been strained by the rise of Google's Android mobile operating system, now the world's leading platform for smartphones.
Google Executive Chairman Eric Schmidt said earlier this week that it has not submitted a new Google Maps app to Apple, but added the search giant talks to the Cupertino-based company "every day."