"I've had a dream for 27 years. I know it's a hell of a long time, but it's finally come true."
Mr Hodgson is one of the longest-serving victims of a miscarriage of justice in the UK.
His solicitor Julian Young said: "I hope that the inquiry, that will undoubtedly take place, will find out how this happened and ensure it does not happen again."
Lord Chief Justice, Lord Judge, and two other senior judges ruled that his 1982 conviction was "unsafe".
The body of Miss De Simone, who had also been raped, was found partially clothed in the back seat of her Ford Escort in the car park beneath the Tom Tackle pub where she worked part-time.
In his ruling, Lord Judge said: "The conviction will be quashed for the simple reason that advances in the science of DNA, long after the end of the trial, have proved a fact which... [would] have resulted in quite a different investigation and a completely different trial."
He said that tests on sperm found at the scene proved it did not come from Mr Hodgson.
"The Crown's case was that whoever raped her also killed her, so the new DNA evidence has demolished the case for the prosecution," Lord Judge added.
He announced at the end of his judgement that Mr Hodgson would be "discharged" and there would be no new trial.
Mr Hodgson said he felt "ecstatic" that he was a free man again
At the time of Mr Hodgson's trial, DNA tests were not available, with the first use of such evidence in court not taking place until 1986 in Leicester.
Mr Hodgson, who is also known as Robert Graham Hodgson and is originally from Tow Law in County Durham, made various confessions to police before pleading not guilty at his trial at Winchester Crown Court.
But his defence said he was a pathological liar and the confessions were untrue.
The prosecution had also been supported during the trial by the fact that blood type analysis available at the time showed that material recovered at the scene belonged to a man with blood of either group A or AB.
Mr Hodgson was in that category along with roughly a third of the male population.
Hampshire Police and the Forensic Science Service undertook a comprehensive forensic case review in November 2008 after requests from Mr Hodgson's legal team.
Police vow to pursue the investigation
It discovered that DNA evidence found at the scene did not match a sample given by Mr Hodgson.
In the light of new evidence, the Criminal Cases Review Commission (CCRC) decided to refer Mr Hodgson's case to the Court of Appeal because it considered that there was a "real possibility that the court would consider the conviction unsafe and quash it".
Det Ch Insp Philip McTavish, from Hampshire Police, said: "Mr Hodgson was convicted by a jury on evidence which included his own admissions to a clergymen, prison officers and police. There were also a number of other strands.
"Hampshire Constabulary has consequently started a reinvestigation into the murder of Teresa De Simone and this is aimed at identifying the owner of the DNA profile.
"We are fully committed to pursuing this investigation."
The Crown Prosecution Service did not oppose the appeal.
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