Hosting Estonia in Stockholm on 27th February, Sweden led by one point going into the final quarter, only to give up 31 points in the final 10 minutes and succumb to an 88-77 loss which added to defeats late last year to Czech Republic and Slovenia.
That meant Sweden - without Miami Heat star Pelle Larsson - had to go to Estonia and beat the same opponents to have any chance of reaching the Second Round, and with only eight minutes to play, the sides were locked together at 58-58.
But this time, it was Sweden who pulled clear in the latter stages, as Melwin Pantzar, Wilhelm Falk and Ludde Hakanson all made telling contributions to a 14-4 run that swung the tie in Blagult's favour.
A three from Hakanson all but ended Estonia's chances of a late rally in the final minutes, as he top scored with 18 points and Sweden ran out 79-69 winners.
Sweden remain bottom of the group, but now the fifth round of matches will be crucial, as Sweden host Czech Republic and Estonia welcome leaders Slovenia.
Should Sweden and Slovenia win their respective games, the Swedes, Czechs and Estonians will all be level on six points with one game to play each.