Bjorkman won 3-6, 6-0, 3-6, 6-2, 6-2 in two hours and 28 minutes in his home town in southern Sweden to earn a home tie against Italy - who beat Spain in their World Group match - a tie which will probably be indoors in September. Sweden, beaten by France in last year's final, have won the Davis Cup five times in 10 finals since 1975.
Bjorkman lost his opening singles match against Wayne Ferreira but teamed up with Nicklas Kulti to win Saturday's doubles against David Adams and Ellis Ferreira.
Wayne Ferreira, ranked 10th in the world, beat Enqvist, ranked seventh, 6-4, 6-4, 6-4 in the first reverse singles to make it 2-2 but Bjorkman was Sweden's hero.
Australia completed a 5-0 demolition of the Czech Republic in their Davis Cup quarter-final tie in Adelaide when Mark Philippoussis and Pat Rafter won their reverse singles matches.
Philippoussis, the Australian No 1, overpowered Martin Damm 6-4, 6-2 in the first of the reverse singles before Rafter beat David Rikl 7-6, 0-6, 6-2.
Philippoussis needed just 66 minutes to beat Damm on the Memorial Drive grass court while Rafter fought back after a sluggish start to overcome Rikl, a lowly ranked player better known for his doubles play.
The tie was effectively over on Saturday when the best doubles partnership in the world, Todd Woodbridge and Mark Woodforde, beat Damm and Rikl 4-6, 6-1, 7-5, 6-4 to give the hosts an unbeatable 3-0 lead.
Australia now face a semi-final against the United States, who beat the Netherlands 4-1 in their quarter-final at Newport Beach in California.
Andre Agassi roared back from a two-sets deficit to beat Jan Siemerink 3-6, 3-6, 6-3, 6-3, 6-3 and then, with the match already decided, Jonathan Stark beat Sjeng Schalken 6-4 6-0.
Siemerink's big lefthanded serve-and-volley style brough him single breaks in the first two sets to give him the chance of rescing his side's challenge but, as in his collapse against Jim Courier on Friday, the Dutchman could not hold on, as Agassi's baseline game finally came to life.