Australia, Germany, Serbia and Italy have won their Davis Cup ties to make it through to the 18-nation World Cup-style event in Madrid in November.
Belgium, Canada, Chile, Colombia, Japan, Kazakhstan, Netherlands and Russia also won their qualifying ties.
The winners join automatic qualifiers Argentina, Croatia, France, Great Britain, Spain and USA in the finals.
Serbia, without world number one Novak Djokovic, were given a scare by Uzbekistan but emerged 3-2 victors.
Uzbekistan were 2-0 down overnight only to level at 2-2, with Denis Istomin partnering Sanjar Fayziev to a 2-6 6-1 6-3 win over Nikola Milojevic and Viktor Troicki in the doubles before Istomin beat Dusan Lajovic 6-3 6-4 in the singles.
However, Serbia went through after Filip Krajinovic came back from a set down to overcome Fayziev 4-6 6-3 6-0 in the final rubber.
Japan had to come back from 2-1 down against China to reach the finals, with Taro Daniel beating Zhe Li 6-3 6-7 (4-7) 6-3 in the deciding match.
Felix Auger-Aliassime beat Norbert Gombos 6-3 6-4 in a decider to send Canada through at the expense of Slovakia, while Chile knocked out Austria thanks to Christian Garin’s 6-2 6-1 victory over Jurij Rodionov in the final rubber.
Switzerland, who were missing 20-time Grand Slam champion Roger Federer and Stan Wawrinka, fell 3-1 to Russia.
There were also 3-1 wins for Italy over India, Kazakhstan against Portugal and for Netherlands in their tie with Czech Republic.
Germany were emphatic 5-0 winners against Hungary, while Australia and Colombia were comfortable 4-0 victors against Bosnia & Herzegovina and Sweden respectively.
The Davis Cup format has undergone an overhaul and has been replaced by the event to be held in Madrid later this year, with matches to be played over the best of three sets across two days rather than three.
Australia 4-0 Bosnia & Herzegovina
Austria 2-3 Chile
Brazil 1-3 Belgium
China 2-3 Japan
Colombia 4-0 Sweden
Czech Republic 1-3 Netherlands
Germany 5-0 Hungary
India 1-3 Italy
Kazakhstan 3-1 Portugal
Slovakia 2-3 Canada
Switzerland 1-3 Russia
Uzbekistan 2-3 Serbia