The visitors were without their top two players through injury, leaving 372nd-ranked Kajsa Rinaldo Persson as their number one, but she overcame a rankings gap of nearly 300 places to defeat Burrage 6-4 6-1 in the opening rubber of the play-off tie.
Burrage had led 4-0 in the first set but from there completely collapsed, losing eight games in a row and 12 of the next 13 as Sweden took a 1-0 lead.
Burrage was unhappy to miss out on selection for the last two teams and had more than justified her debut with a strong season, breaking into the top 100 for the first time and establishing herself as the clear British number two behind Katie Boulter.
She could not have asked for a better start, saving break points in the opening game and then breaking serve twice in a row.
But Burrage’s confidence proved fragile and the occasion began to overwhelm her as Persson worked her way calmly back into the set.
Routine shots from the 24-year-old were flying yards long and her unforced error count hit 23 as a sixth game in a row went the way of Persson.
The capacity crowd of 5,500 tried to lift her in the second set and she finally stopped the run of games against her to make it 1-2, but she could not mount a comeback, with Persson now playing well above her lowly ranking.
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