Appeal judges concluded that there was "overwhelming evidence" Colin Coats and Philip Wade were guilty of the crime.
Coats, 43, was sentenced to a minimum of 33 years in April last year after a judge concluded he was the ''prime mover'' in the abduction, torture and killing of the 27-year-old.
Wade, also 43, was ordered to spend at least 30 years behind bars after being convicted of the same charge at the High Court in Glasgow.
Ms Spence was reported missing in April 2011 and her body has never been found.
Both men launched appeals claiming that they were the victims of a miscarriage of justice.
Their legal teams said trial judge Lord Pentland was wrong to dismiss an objection to a line of evidence during the trial.
The evidence involved a statement made by Wade that he had recently killed someone and needed to find a way to dispose of the body.
In his report on the trial, Lord Pentland explained his reason for the objection, saying: "I considered that the evidence was obviously relevant to proof of the charges on the indictment and that the objection to its admissibility was misconceived.
"The evidence had been disclosed well before the trial and it was not disputed that Wade had been given advance notice of it."
The appeal judges - Lord Justice General, Lord Gill, sitting with Lord Menzies and Lord Turnbull - agreed he had been entitled to refuse the objection.
Lord Gill said: "I have no reason to think that his refusal was unreasonable or perverse.
"The objection could reasonably have been raised before the trial."
Lawyers for Wade and Coats also argued that the Crown had not properly disclosed the fact that Ms Spence had been recruited as a covert intelligence source by the Scottish Crime and Drug Enforcement Agency (SCDEA) a month before she disappeared.
This had prejudiced the defence and led to a miscarriage of justice, they told a hearing at the Court of Criminal Appeal in Edinburgh.
But Lord Gill said: "There emerged, in my view, an overwhelming case that the appellants murdered the deceased. In my opinion, there was no miscarriage of justice."
On the sentencing appeal, he added: "This was a barbaric crime that merited the severest penalty.
"The sentencing judge, who heard all of the evidence, considered that these sentences were appropriate. I see no basis on which we should interfere with them."
Following an 11-week-trial David Parker, 38, and Paul Smith, 47, who admitted holding Ms Spence at Parker's flat in West Kilbride, Ayrshire, were also sentenced to 11 years and three months, and 11 years respectively.
They were originally charged with murder but their pleas of guilty to a reduced charge of detaining Ms Spence against her will, assaulting her and attempting to defeat the ends of justice were accepted by the Crown.
Jurors heard how Ms Spence was forced into a car in Broomhill Path, Glasgow, on April 14 2011 and driven to Parker's property in Meadowfoot Road.
She was taped to a leather chair by the arms and waist, and tortured as Coats and Wade tried to extract financial information from her.
The court heard how she orchestrated an apparent land deal at Stansted Airport in Essex which was supposed to be ''worth millions'' to Coats.
Ms Spence was burned with an iron, hit with a golf club, had her toes crushed, thumb cut off and little finger severed before finally being murdered and having her body disposed of, the court was told.