WASHINGTON—The Democratic National Committee’s computer systems were compromised by hackers linked to the Russian government, in one of the largest known breaches of a U.S. political organization, party officials said Tuesday.

Two groups with ties to the Russian intelligence services hacked the party’s information-technology infrastructure, said CrowdStrike Inc., a security firm hired by the Democratic Party after the intrusion was discovered. The same groups had previously penetrated unclassified networks at the White House,...