Sweden stayed in contention in Boras after Johan Brunstrom and Robert Lindstedt scored a vital win against Thomas Kromann and Frederik Nielsen in the doubles on Saturday.
The Swedish pair went into the match knowing that only a win would suffice after the Danes had taken a 2-0 lead on the opening day.
After dropping the first set in a tiebreak, Brunstrom and Lindstedt fought back to oust the opposition 67(5) 64 63 63 in two hours 34 minutes.
Meanwhile in Moscow, the Russian doubles duo of Konstantin Kravchuk and Andrey Kuznetsov overcame South Africa's Raven Klaasen and Tucker Vorster 26 63 76(4) 62 to clinch the tie in clinical fashion.
Russia take an unassailable 3-0 lead into the final day at the Olympic Stadium and in so doing have secured their place in Europe/Africa Zone Group I for another year.
By losing the 2nd round relegation play-off, South Africa have been relegated to Europe/Africa Zone Group II in 2014.