WASHINGTON—The nation’s commercial banks and savings institutions reported a 13% rise in net income in the third quarter, hitting a record as institutions’ profits soared and expenses moderated.

Net income at the 5,980 banks insured by the Federal Deposit Insurance Corp. rose $5.2 billion, to $45.6 billion, in the third quarter, compared with a year earlier, according to data released Tuesday by the FDIC.

“The...