Charles Owenson, 62, and James Costello, 44, received bribes  from ABC Ltd staff members Kevin Balmer, 52, and Brendan Cantwell, 44.
For cash, the council workers helped award contracts to the Edinburgh firm.
The charges related to the maintenance of Edinburgh Council buildings between 2006 and 2010.
The council workers pleaded guilty on Tuesday at Edinburgh Sheriff Court to breaching the 1889 Public Bodies Corrupt Practices Act.
The men also admitted committing fraud and breaching proceeds of crime charges at a hearing before Sheriff Michael O'Grady.
During the hearing, depute procurator fiscal Keith O'Mahoney told the court the scam was so complicated he would need another day in court to explain how it worked.
However, the contents of a legal document given to the press detailed the nature of the charges against Owenson and Costello, both of Edinburgh, and Balmer and Cantwell, both of Livingston, West Lothian.
The first charge details how between 6 January 2006 and November 10 2010, Balmer and Cantwell corruptly gave £42,521 to Owenson and Costello at various locations in Edinburgh.
The two ABC Ltd employees also gave Owenson and Costello £30,249 in hospitality as an inducement to secure contracts for the repair and maintenance of Edinburgh City Council buildings.
The second charge states Owenson acted corruptly whilst working as an officer at the local authority's Property Care Services Ltd.
It states that he received £28,387 at various locations in Edinburgh between 2006 and 2010 as a reward for the council awarding contracts to ABC Ltd.
The third charge states Costello acted corruptly whilst acting as an officer in the same department and received £14,134 as a reward for the council awarding contracts to ABC Ltd.
Between 18 April 2007 and May 2010, at various locations in Edinburgh, Balmer fraudulently caused Edinburgh City Council to pay ABC Ltd £67,923 for work the company carried out on behalf of the local authority.
Balmer took £22,641 from the fraudulent scheme.
Between 19 May 2007 and 29 December 2011, at various locations in Scotland, Owenson breached Proceeds of Crime legislation by laundering £28,387 by purchasing cars.
Costello carried out the same scam between June 2007 and April 2012 at various locations in Scotland.
Sheriff O'Grady adjourned the case until Friday.

A 19-year-old man was shot in the leg in Stainbeck Road on Thursday.
He remains in hospital, where his condition is described as stable, West Yorkshire Police said.
A 29-year-old man, from Woodhouse, is in custody. The force said extra officers were patrolling the area to "provide reassurance to the local community" while enquiries continued.

Medical sources told the BBC that at least eight people, including civilians, have been killed and more than 20 injured.
A new UN-backed Government of National Accord (GNA) still lacks authority and the country remains volatile.
The French Foreign Ministry said it was "very worried by the escalation of violence".
The BBC's North Africa Correspondent, Rana Jawad, in neighbouring Tunis, said two of the city's largest and most heavily armed militias are fighting an alliance of hard-line Islamists and militias loyal to rival political authorities.
Libya is still politically divided as rival administrations attempt to assert their power.
Clashes are said to be continuing in two main districts in the south of the capital. Tanks and pickup trucks armed with rocket launchers were reportedly deployed in several districts.
Witnesses told the BBC that people stayed at home on Friday morning and many businesses stayed closed.
The new government in Tripoli has not commented on the clashes.
East and central:
West:
Guide to Libya's militias
Profile: Khalifa Haftar
Rogue general divides Libyans

At least five vehicles are believed to have been involved in the collision at J18 Charing Cross.
Emergency services are at the scene and traffic is being directed off at J17, down Newton St and back on at J19, before the Kingston Bridge.
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Two Edinburgh City Council employees have pleaded guilty to receiving tens of thousands of pounds worth of bribes from building company workers.
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