Kyriakos Amiridis had been missing since Monday. His body was found in a burnt-out car on the outskirts of Rio de Janeiro on Thursday.
Police say the ambassador's Brazilian wife, Francoise Amiridis, plotted the murder with her lover.
Sergio Gomes Moreira Filho confessed to strangling him to death, police said.
Mrs Amiridis, 40, has been detained along with Mr Moreira, 29, and his cousin, Eduardo Melo.
Mr Melo is said to have been paid about 80,000 reais ($25,000; Â£20,000) to act as a lookout.
The three suspects have been questioned and will be held in police custody for another 30 days.
Investigator Evaristo Pontes Magalhaes described the murder as "a tragic, cowardly act," which police are treating as "a crime of passion".
"Francoise initially denied the facts. She said she had nothing to do with that. We managed to make her see that she had no alternative and that there was no point in continuing to deny it," said Mr Magalhaes at a news conference.
"She fell into contradictions, burst into tears and began to say that the police officer [Sergio Moreira] had carried out her husband's murder," he added.
Mr Magalhaes said Mr Moreira had a fight with the ambassador and ended up killing him when he went to confront the diplomat about claims of violence against his wife.
He said he acted in self defence. But Brazilian police have dismissed his statement.
They say the policeman's cousin confessed to moving the ambassador's body and gave full details of the plot to murder Mr Amiridis, implicating both Mr Moreira and Mrs Amiridis.
The 59-year-old ambassador had travelled from Brasilia to the city of Nova Iguacu, north of Rio, to spend the Christmas and new year holidays there with his wife and her parents.
The couple had been living together for 15 years and their daughter is 10 years old, local reports said.
Mrs Amiridis reported her husband missing on Wednesday.
She told police that he had left on Monday in a car he had rented. The burnt-out vehicle was found on Thursday under a flyover on one of the main access roads to Rio with a body inside.
The body was burned beyond recognition, but police said it was that of Mr Amiridis.
Investigators also said that they had found blood stains on the sofa in the house where the couple was staying and say he was probably stabbed to death there.
Mr Amiridis served as consul in Rio de Janeiro between 2001 and 2004 and returned to the country as ambassador earlier this year.

Pupils at The Joseph Whitaker School in Rainworth, Nottinghamshire, broke the record at their first attempt with a speed of 478.1mph (769.4km/h).
However, later, a different car they had built went even faster and hit an average speed of 533.1mph (857.9km/h).
The footage from the Rolls-Royce runway in Hucknall will now be sent to Guinness for verification.
If the speed is verified, it will have smashed the existing record held by Samvir Thandi  and the Heathland School in Middlesex, whose car reached 287.6mph (462.84 km/h) in March.
The Young Engineers tweeted: "We have a confirmed 478.1mph average by Red Shift! Well done to all the young engineers - world record holders!!"
However, after a further attempt with a car they had nicknamed Insanity, the team went on to set an even bigger record.
Phil Worsley, the teacher in charge of the Young Engineers club, said: "When people watch our rockets they don't anticipate the speed, their heads lag behind the car.
"[The pupils] are absolutely amazing. The most amazing bunch of people I've ever worked with."
The cars, which are about 45cm (17in) long and constructed from carbon fibre, run on titanium wheels and are guided by a 250m (820ft) steel cable.
All parts have been stuck together with strong glue and at the end of each run, the cars come to a stop using small parachutes.
The pupils have been working on the project for about a year.
Police in Brazil say the Greek ambassador to the country was killed by a local police officer who was having an affair with the envoy's wife.
[NEXT_CONCEPT]
A team of schoolchildren's rocket-powered model car has broken the world speed record, it is claimed.