John Pottinger, 21, admitted causing the death of 39-year-old George Armour by dangerous driving, when he appeared at the High Court in Glasgow.
The incident happened on the B9074 at Hamnavoe in June last year. The victim was thrown up and over the bonnet of the car.
Sentence was deferred on Pottinger until next month.
The court heard Pottinger was "pretty drunk" after attending a disco at the Burra Hall.
Advocate depute Paul Brown, prosecuting, told the court: "The disco finished at around 01:00 and people from the hall made their way outside.
"Amongst those leaving the disco was the now deceased Mr Armour who had attended the disco with members of his family."
Pottinger decided to go to a party on a nearby island, and was persuaded to drive his Volkswagen Golf, taking four teenage friends with him.
Mr Brown added: "The accused was then seen by a witness to be sitting in the driver's seat of his vehicle drinking and trying to start his car.
"He managed to moved the vehicle a short distance partially on to the road but stalled in the middle of the carriageway.
"He tried to restart the car and stalled around eight times."
An onlooker drove the car a short distance for him before he got back in and made off in the direction of Scalloway.
He drove off with "smoke billowing" and bouncing his car over speed bumps.
Two of his friends told him he was going too fast and shouted at him to slow down as they approached the brow of a hill immediately before the collision.
Mr Brown told the court: "The front seat passenger said that the vehicle at this point bumped up on the left-hand verge and he could see a man, who was the now deceased Mr Armour, walking on that verge away from them.
"The car continued on the verge and struck Mr Armour who was thrown up and over the bonnet of the vehicle."
Pottinger did not stop and continued down an embankment and into a field and, despite cries to stop, kept driving.
It was only when one of his friends pulled the handbrake that the car stopped. The passengers got out the car and ran back to Mr Armour, but Pottinger drove off.
When police later approached him, Pottinger asked them: "Just tell me is he dead?"

The detonation happened shortly before 22:00 local time (20:00 GMT) in the Schwabing district and was heard across the city, local media report.
There are reports that sparks from the explosion caused the roofs of some neighbouring buildings to catch fire.
The bomb was discovered on Monday night by building workers at the site of an old bar that was being demolished.
Overnight, 2,500 residents were evacuated from the area closest to the bomb, with others living further away being told to stay in their homes.
Experts decided it was not possible to make the device safe because of its unusual fuse, which operated by means of a chemical reaction rather than the mechanical device that many Allied World War II bombs used.
The bomb was described as a highly explosive, a 550lb (250kg) device dropped by the Americans.
It is not unusual for big, unexploded bombs to be discovered in Germany, the BBC's Stephen Evans reports.
About 600 tonnes of unexploded ordnance are discovered in Germany every year.
They are usually defused safely, though in 2010 three bomb-disposal officers were killed during an attempt to make a bomb safe, our correspondent adds.
In December 2011, the biggest bomb disposal operation in Germany since 1945 was mounted in Koblenz to defuse two bombs from World War II found in the riverbed of the Rhine.
A car driver has admitted knocking down and killing a man in Shetland after a day of drinking.
[NEXT_CONCEPT]
A bomb disposal team has detonated an American bomb left over from World War II found in the German city of Munich.