Israel’s Harel Levy sealed the crucial point by beating Andreas Vinciguerra 6-4, 4-6, 6-4, 3-6, 8-6 in just over three and a half hours after Israel had started the final day of the World Group first round tie 2-1 down.
Levy, ranked No. 241 in the world, has never lost a five-set match and has now extended his winning record to 5-0.
The 30-year-old clinched victory on his second match point, after brave resistance from Vinciguerra, who has returned to the game after a two-year injury absence.
Israel returned to the World Group last year after a 14-year absence, only to lose to Sweden in the first round.
Having now avenged this loss, they can now look ahead to a home tie against Russia in July in what will be their first quarter-final since 1987.
Sunday’s defeat represented the first time in Sweden’s 84-year Davis Cup history that they had lost a tie from 2-1 up.
Earlier in the day, Dudi Sela had pulled Israel level at 2-2 by beating Thomas Johansson 3-6, 6-1, 4-6, 6-4, 6-2.
The match had been played behind closed doors as protestors, demonstrating against Israel’s bloody December Gaza offensive, clashed with police outside.
Israel captain Eyal Ran, who celebrated the birth of his first son last weekend, tried to forget the furor over the protests which had been aimed at trying to get the match called-off.
“I think tennis was the winner of this weekend. A fantastic sports event and great matches,” said Ran.
“We made history this weekend and I hope people at home appreciate our win over a fantastic Davis Cup country. We are also looking forward to a quarter-final against Russia.”
Levy said the absence of a crowd at the Baltic Hall could have worked against the Swedes.
“Playing two matches this weekend with 8-6 scores in the fifth. It was so close, just a point here and there,” said Levy.
“I feel sorry for Vinciguerra and maybe Sweden lost the tie because they didn’t have the normal advantage of a home crowd.”