Hackers breached the payment systems of Saks Fifth Avenue and Lord & Taylor department stores and stole credit card information for millions of shoppers, the latest in a series of intrusions that have exposed security gaps in corporate networks.

Hackers claim they have five million credit card and debit card numbers from the stores and have been releasing them for sale on the “dark web,” a network of websites used by hackers and others to anonymously share information, according to Gemini Advisory LLC, a New York-based cybersecurity...