Relations between the two countries are at their worst in years, with each accusing the other of subverting regional security and support opposite sides in conflicts in Syria, Yemen and Iraq.
"This was one of three vessels which were intercepted by Saudi forces. It was captured with the three men on board, the other two escaped," a statement from the ministry's center for international communications said.
"The three captured members of the Iranian Revolutionary Guard are now being questioned by Saudi authorities," it said, citing a Saudi official.
The vessel, seized last Friday, was carrying explosives and intended to conduct a "terrorist act" in Saudi territorial waters, it said.
An earlier report from the Saudi Press Agency said the Saudi navy had fired warning shots at the two boats that managed to escape.
Iran's Tasnim news agency said on Saturday that Saudi border guards had opened fire on an Iranian fishing boat in the Gulf on Friday, killing a fisherman. It said the boat was one of two Iranian boats fishing in the Gulf that had been pushed off course by waves.
Tensions between Iran and Saudi Arabia have steadily deteriorated. On June 5, Riyadh and other Arab governments severed ties with Qatar, citing its support of Iran as a reason.
Days later, suicide bombings and shootings in Tehran killed at least 17 people. Shi'ite Muslim Iran repeated accusations that Saudi Arabia funds Sunni Islamist militants, including Islamic State. Riyadh has denied involvement in the attacks.
(By Stephen Kalin; Additional reporting by Hadeel Al Sayegh; Editing by William Maclean, Robert Birsel, Richard Balmforth)