Britain, appearing at the team event for the sport's top eight nations for the first time in 20 years, needed to win the tie to stand a chance of reaching tomorrow's final.
Bjorkman used blistering returns and heavy groundstrokes to crush Rusedski in just 51 minutes. The British world No 13 struggled to reproduce his usually devastating serving and made several unforced errors during a one-sided match.
Former top 10 player Bjorkman, currently 28th in the ATP Tour rankings, originally arrived here to play doubles but was picked for singles as well after Thomas Johansson hurt his left wrist in practice on Wednesday.
"I made him play badly." Bjorkman said. "I managed to return his serve and against Greg, that's the key. I went into the match with a lot of self-confidence since I had already beaten him five times."
Henman, the world No 7, who had won his previous two matches against Enqvist but had never played him on clay, put up a brave challenge but the Swede showed more composure when it mattered most.
Sweden ruined the hopes not only of Britain, but also of Spain, who were seeking a record fifth triumph in the tournament. They won their tie against Germany 2-1, but cannot qualify for the final after Britain's exit.
Alex Corretja set Spain on their way by beating Nicolas Kiefer 6-3, 4- 6, 6-4, and Carlos Moya sealed the result with a straight-sets victory over Tommy Haas. Boris Becker and David Prinosil won their doubles tie against Albert Costa and Jairo Velasco, but it was small consolation for the home fans. Sweden now meet the winners of today's match between Australia and the United States.
Corretja, the world No 6, said he was pleased with his last warm-up for next week's French Open. "To win a long, tough match is the best preparation you can get," he said. "I'm feeling good and I'm confident."
Kiefer delighted the home fans by fighting back from 4-2 down to win the second set. But Corretja was back in command in the final set. "I can live with that defeat," the German said. "I lost to one of the best clay-court players in the world and it was close."
n Boris Becker will begin his grass-court preparation for his final appearance at Wimbledon by playing exhibition matches during the Powder Byrne Trophy at Surbiton (29 May to 6 June). Meanwhile Jimmy Connors is to join his fellow former Wimbledon champions Bjorn Borg and John McEnroe at the Honda Challenge, the London leg of the Delta Airlines ATP Senior Tour of Champions, which returns to the Royal Albert Hall on 1 December.