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.
Visitors to the exhibition were free to use the palace's throne for its intended purpose, with a three-minute time limit to avoid queues.
Det Insp Jess Milne, said: "The piece of art that has been stolen is a high-value toilet made out of gold that was on display at the palace.
"We believe a group of offenders used at least two vehicles during the offence.
"The artwork has not been recovered at this time but we are conducting a thorough investigation to find it and bring those responsible to justice."
In a tweet, Blenheim Palace said it would remain shut for the rest of the day, but would reopen on Sunday.
Palace chief executive Dominic Hare said they were "saddened by this extraordinary event, but also relieved no-one was hurt".
"We hope that the wonderful work of our dear friend Maurizio Cattelan becomes immortalised by this stupid and pointless act," he added.
The gold toilet was famously offered to US President Donald Trump in 2017.
The arrested man is in police custody.
Valuable and historic antiques stolen as thieves raid Gloucestershire Castle
Gloucestershire Constabulary have revealed details of a burglary committed at the Winchcombe stately home earlier this month.
In the early hours of Sunday, September 8, offenders forced their way into the exhibition area of the castle, smashed a display case and stole valuable jewellery and artefacts.
In a statement released today, Police have revealed that the offenders arrived and departed in a 4x4 vehicle with a number of items - including a chair.
The statement said: "The detectives are examining CCTV footage which shows four offenders approach the property with a chair, sledgehammer and large garden or builder's bag."They made off in a 4x4 vehicle which was parked nearby."
The stolen property includes:
- A presentation gold box, Swiss late 18th Century, with a miniature of Edward VII set in diamonds;
- A presentation gold and enamel box with the Prince of Wales feathers set in diamonds;
- A Cartier watch monogrammed AK (Alice Keppel) 1910; a Faberge gold cigarette case set with diamonds;
- A Faberge silver caviar box and silver mounted vodka glasses, c1910;
- A gold snuffbox, Paris 1783;
- A silver cigarette case by Marshak, Kiev 1908;
- and a book of extracts from Sydney Smith, given to Alice Keppel by King Edward VII.
"Clearly the stolen items are very distinctive and have a great deal of historical, as well as financial, value.
"They should be easy to identify and if anyone is aware of them being offered for sale I would urge them to report it to police as soon as possible".
Sudeley Castle's owner Lady Ashcombe said: "We are all saddened to learn of this burglary. There were beautiful artefacts on display for everyone to enjoy and were very precious to me personally".
Anyone with information about the burglary is asked to call Gloucestershire Constabulary on 101 quoting incident 110 of 8 September.
Thieves nab €2 million haul from vaunted French chateau
The Vaux-le-Vicomte chateau, set amid sumptuous gardens about 50 kilometres southeast of Paris, has been owned by the same family since 1875.
A police source said six hooded but unarmed robbers found neckties belonging to Patrice de Vogue, 90, to tie up him and his wife Cristina, 78, in their lodging on the grounds of the palace.
The thieves targeted a safe and stole emeralds and other items, but did not attempt to make off with the chateau's works of art.
The couple was not injured during the robbery, which occurred shortly before dawn, according to prosecutors in the nearby city of Melun.
"The owners are doing fine and the chateau remains open for visits as usual," the palace's management told AFP.
Patrice de Vogue opened the estate to the public in 1968, and it is now run by the couple's three sons.
The chateau was built by Louis XIV's finance minister Nicolas Fouquet, who according to legend fell from grace in 1661, shortly after the building work was finished, when he staged an elaborate party there and aroused the Sun King's envy.
Fouquet spent the rest of his life in jail. The monarch then seized the palace and moved its most precious artworks and other objects to Versailles.
It is the largest privately-owned heritage site in France, sprawled over 500 hectares, and has some 250,000 visitors each year.
The chateau often stands in for Versailles for movie and television productions, from the Roger Moore Bond hit "Moonraker" to Sofia Coppola's "Marie Antoinette".
It has become a prize spot for celebrity wedding parties, such as the lavish 2007 bash for French basketball star Tony Parker and "Desperate Housewives" star Eva Longoria.