The first two sets went at breakneck speed and Sweden, with recently joined star and recent Champions League champion Isabelle Haak, looked to take an easy victory.
But Slovenia, who won against Luxembourg in their first match in the league, rose in the third set and even took the lead with 14-13, before Sweden turned and also won the third set.
The whole match was decided in just over an hour.
Next, Sweden will meet Luxembourg on 1 June.
Then Portugal and Estonia wait, before the return meetings between the teams start.
The final will be played on July 2.
CUT: That's why the compass is so important
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That's why the compass is so important