Voskamp, whose team leader Jeroen Blijlevens won the fifth stage, beat Germany's Christian Henn in a sprint finish with Alberto Elli of Italy in third over 20 seconds behind.
Denmark's Bjarne Riis retained the overall leader's yellow jersey finishing in the peloton nearly 17 minutes behind the winner. "It's amazing as I thought I would never get over the Pyrenees but now I have capped that by winning the stage," Voskamp said.
The 27-year-old Dutchman, whose only other major triumph was a 1994 stage of the Tour of Spain, was part of a 14-man breakaway group that broke away on the fourth climb of the stage, the second category D'Ispeguy.
Among others in the group were Australian Patrick Jonker, more than 33 minutes behind Riis, permanent escapee Alberto Elli of Italy, Pascal Herve and Bruno Boscardin, from King of the Mountains winner Richard Virenque's Festina team, and Eric Breukink, third overall in 1989.
The group was further split up after they had built up an unassailable lead, with the peloton taking it easy after the rigours of the Pyrenees, with 14km to the finish.
Voskamp, Henn, the German national champion and workaholic teammate of Riis, Bruno Thibout, riding for the Motorola team whose Italian cyclist Fabio Casartelli was killed on the Tour exactly a year ago, Boscardin and Elli escaped from the other nine.
Voskamp refused to lead the relay, but he was the only one able to lock onto Henn's wheel when the veteran fled with eight km to go.
Thibout expressed his frustration at the finish.
"I told Boscardin to help track back to them but he refused and that was that. You would have thought on the day that Fabio was killed he would have helped one of his former teammates to win, alas that was not the case," Thibout sighed.
Thibout also missed out when Elli beat him in the sprint for third, adding to the frustrations of the Motorola team, which lost team leader Lance Armstrong early on in the race.
Before the start of the day's 154.5-km ride from Pamplona, Spain, won by Dutchman Bart Voskamp, the best young rider of this year's Tour, German Jan Ullrich, offered a white flower to each rider of the Motorola stable.
Olympic road race champion Casartelli, who was with Motorola, was killed at the age of 25 in a crash in the descent of the Portet d'Aspet mountain pass, in the Pyrenees, on July 18 last year.
On Tuesday, Tour director Jean-Marie Leblanc, accompanied by Motorola rider Frankie Andreu and officials of the American team, paid him homage by going to the white marble monument built by the side of the road where he fell.
His widow Anna-Lisa will be in Paris where the race ends on Sunday to give his reward to the winner of the best young rider's challenge, now named after Casartelli.
The riders must have had him in mind when they progressed slowly on Thursday's stage to the port of Hendaye, on the French Basque coast, covering only 27 kms in the first hour.
Eleven men later broke away up the penultimate climb of this year's Tour, the Ispeguy pass, 80 kms into the stage.
The peloton failed to react and the men in front increased their lead in the last climb, the Puerto-Otxondo pass, just before the French border.
Some dropped back and there were only five riders left in the leading group in the last 20 kms. Then, with eight kilometres left, Voskamp and German champion Christian Henn escaped.
The 28-year-old Voskamp outsprinted Henn to claim his first Tour stage and Italian Alberto Elli crossed the line a few seconds later to take third place.
The pack, including Riis, came in nearly 17 minutes later at the end of a stage which brought no changes to the leading overall standings.
Riis still holds a comfortable lead of three minutes and 59 seconds on second-placed Ullrich with Frenchman Richard Virenque third a further 26 seconds back.
The 32-year-old has little to fear from the the three remaining stages and should become the first Dane to win the Tour after Sunday's traditional finale on the Champs-Elysees.
Voskamp, who normally concentrates on preparing the sprints for fellow Dutchman Jeroen Blijlevens, his team mate in the TVM stable, seized his chance to shine.
``Usually, I have to work for Jeroen but as I was in a breakaway group and it was obvious that there would not be a sprint, I had an opportunity,'' said the Dutch rider, whose previous significant success was a stage of the 1994 Tour of Spain.
Voskamp might have to work again in Friday's 19th stage, a flat, 226.5 km ride to Bordeaux which could well end with a mass sprint.
They were protesting at the Tour organisers lenient treatment of the Kelme team boss Alvaro Pino, who was given a severe warning by Leblanc, after he knocked two employees of the French sports paper L'Equipe off their motorbike during Wednesday's 17th stage.
Pino did not stop to see how they were and the two men were taken to hospital with numerous cuts and bruises.
Leblanc did not condone his behaviour but said that he could not suspend him as the photographers and motorcyclists wanted him to. "I understand their feelings which could have had serious consequenses on a steep descent, particularly after the riders had just been obstructed by some Basque demonstrators. I demanded eyewitness accounts and explanations from the three parties concerned," Leblanc said.
"Pino should have asked how they were and I am going to ask him to do that. I couldn't ban him as I was not sure he had done it on purpose and the witnesses could not confirm this," Leblanc added.
The photographers and their drivers wanted him banned from the rest of the Tour. They said that as a result of his 'lenient treatment' they would follow the Tour from behind the peloton and not take any pictures.
"The penalties are more severe for us if we touch a cyclist than if a team boss touches a motorcyclist. Its really dangerous to cover the Tour de France and we would expect more protection from the organisers," a L'Equipe photographer said.
"We will provide pictures only of the finish of the 18th stage and the podium," he added.
Roger Legeay, boss of British Olympic and World pursuit champion Chris Boardman's GAN team, criticised Pino's behaviour. "We all take risks and generally things go well but yesterday it was serious and I regret that Pino did not stop to see how they were," Legeay said.
"Regarding the decision of the organisers I have seen several managers such as Jan Raas banned for a day so there is a precedent. I hope Pino will make a public apology to the photographers," Legeay added.
Indurain, who represented Spain at the Los Angeles Olympics in 1984, said he had not decided whether he would contest the time-trial at the Olympics which start on July 20.
"I will see how I do in the time-trial from Bordeaux to Saint Emilion on Saturday and if I feel satisfied then I will go to Atlanta. Otherwise I will go on holiday with my wife and baby Miguelito," Indurain said.