An investigation by Nottinghamshire Police into the attempted robbery led to the series of crimes being linked together.
Two men, including one with a baseball bat, were said to have tried to grab cash from the Post Office in Scrooby Road, Bircotes, at 8.11am yesterday, Saturday, January 22.
One tried to jump over the counter while the other tried to hold a member of staff back - but they were fought off and the suspects left empty-handed.
They then fled in a distinctive black and orange Toyota Aygo.
Officers investigating the incident later linked the vehicle to a theft of fuel at the Co-op petrol station in North Road, Retford, which took place at around 8.25am, where a driver and two passengers were seen in the car.
The three men are also believed to have gone on to attempt to break into a house in Canal Lane, West Stockwith, by throwing a slab through a patio door at some time between 10am and 12.15pm the same day.
Further investigation led officers to suspect the vehicle was one that had been stolen in Doncaster on Friday night.
Officers later found the vehicle in Cumberland Close, Bircotes, and following a search of the area they detained three suspects.
Three men, aged 51, 42 and 34, were each arrested on suspicion of attempted robbery, theft of a motor vehicle and making off without paying.
They remain in custody.
Detective Sergeant Lauren Morgan, of Nottinghamshire Police, said: "This was an excellent example of investigative policing, with officers from our response, neighbourhood policing and CID teams working together to link the offences and get three suspects in custody in a very short space of time.
"Robbery is a serious offence and while the member of staff was not hurt, clearly incidents like this can be unsettling, and we are committed to ensuring that people who target victims in this way are brought to justice.
"Our enquiries are continuing and while we have made three arrests in connection with the offences we would still like to hear from anyone who saw anything suspicious in the area."