The public gallery at Leeds Crown Court erupted as the jury's forewoman delivered the verdicts after more than four hours of deliberation.
Sentencing Grieveson to three life sentences, Mr Justice Holland described him as "plain evil". The judge said he would recommend to the Home Secretary that his successors think "long and hard" as to whether Grieveson was still a risk to the public before releasing him.
Despite an earlier request for people in the gallery to remain calm, there were cheers and sobs of relief.
The jury decided that 25-year-old unemployed Grieveson was responsible for the series of deaths at Sunderland, Tyne and Wear, in circumstances so bizarre, that more than seven months went by before police launched a murder hunt.
Grieveson, formerly of Roker Avenue, Sunderland, had denied the murders of Thomas Kelly, 18, and David Hanson, and David Grieff, both 15, over a three month period.
The bodies of Kelly and Grieff, both from Sunderland, were found in burnt out huts on allotments in the city's Monkwearmouth area on 26 November 1993, and 25 February 1994, respectively.
Firemen discovered the body of Hanson, also from Sunderland, when they tackled a blaze started inside an empty house on the city's seafront on 8 February 1994.
After the verdicts Mr John Milford QC, prosecuting, raised the fact Grieveson was also charged with the attempted murder of a 14-year-old boy, who cannot be named for legal reasons, in the Hendon district of Sunderland, in August 1991.
He said that in view of the verdicts there would be "no profit" in trying that case. It would be left on file.
When the trial opened on 29 January, Mr Milford said Grieveson was a homosexual who was either "unable or unwilling to accept his sexuality", and that he killed the youths for two reasons. One was "to prevent them from revealing that he had demonstrated his sexual preference to them". The other was "simply because he enjoyed killing them and firing their bodies".
The jury heard Grieveson was interviewed by police after each of the killings, but it was not until seven months later that Northumbria police launched the investigation which led to him being charged. This was because initial post-mortem examinations did not indicate the causes of death and it took two of the country's most eminent pathologists to establish all three youths were strangled.
Outside court, Detective Superintendent Dave Wilson, who led the murder inquiry, said: "Grieveson is a very dangerous man who should not be released. He would have gone on to kill again."
The victims' parents, Tommy and Judy Kelly, John and Sheila Hanson, and Janet Grieff and Ray Gilston, were trembling with emotion. Mr Kelly said: "It is a great relief this monster is off the streets so no other family will have to go through what we faced."