A gang broke into the Oxfordshire palace at about 04:50 BST and stole the artwork, Thames Valley Police said.
The working toilet - entitled America, which visitors had been invited to use - has not been found but a 66-year-old man has been arrested.
The burglary caused "significant damage and flooding" because the toilet was plumbed into the building, police said.
It was part of an exhibition by Italian conceptual artist Maurizio Cattelan that opened on Thursday.
The 18th Century stately home is a World Heritage Site and the birthplace of Sir Winston Churchill. It is currently closed while investigations continue.
Speaking last month, Edward Spencer-Churchill - half-brother of the current Duke of Marlborough - said he was relaxed about security for the artwork.
"It's not going to be the easiest thing to nick," he said.
Photo Credit: Jacopo Zotti/MENOMENOPIU ARCHITECTS/Maurizio Cattelan Archive