WASHINGTON—New details of the $400 million U.S. payment to Iran earlier this year depict a tightly scripted exchange specifically timed to the release of several American prisoners held in Iran.

The picture emerged from accounts of U.S. officials and others briefed on the operation: U.S. officials wouldn’t let Iranians take control of the money until a Swiss Air Force plane carrying three freed Americans departed from Tehran on Jan. 17. Once that happened, an Iranian cargo plane was allowed to bring the cash home from a Geneva...