Grzegorz Gamla carried out a frenzied attack on fellow Pole Maciej Ciania, 34, in Dickson Street in January last year.
Gamla later fled back to his homeland before being arrested and brought back to Scotland to face justice.
The 26 year-old was ordered to serve a minimum 19 years behind bars as he appeared at the High Court in Glasgow.
Lord Pentland told Gamla: “It is clear that this was a sustained and savage onslaught pressed home with ruthless determination.”
The judge added the victim’s relatives had suffered “terrible anguish” as a result of the death.
Gamla – who had been convicted after a trial last month - will be deported when he is eventually released.
The court heard how Mr Ciania was stabbed and slashed 37 times after a “simmering dispute” over unpaid household bills.
Gamla soon escaped from the property in the capital’s Leith area after their flatmate discovered the corpse.
Gamla had used power tools to make a box to hide the body.
He initially took a ferry from Stranraer to Belfast before moving onto Dublin and then flying to Warsaw.
Gamla was arrested by the Polish authorities and returned to Scotland on an extradition warrant.
The killer was also found guilty of assaulting Mr Ciania on a previous occasion.
Brian Gilfedder, defending, today said: “It appears there was a love-hate relationship between the two men.
“There certainly was a simmering background, but the accused never believed that it would come to this.”
Mr Ciania’s heartbroken sister told after the verdict how her brother had lived in Edinburgh for 10 years and had “regarded the city as a second home”.
Aleksandra Poprawska added: “Nothing will bring back Maciej, but we are relieved to know that the person who committed this terrible crime will probably never hurt anyone again.”