Kontaveit turned the second set around from 4-0 down and mounted another comeback to win the decider with a break of serve when Alexandrova served for the match at 5-4 up.
The Indians thus finished their group engagements at second place behind China, setting them up against Denmark in the last-eight stage on Friday.
B Sai Praneeth began the proceedings with a 21-5 21-6 win over Louis Beaubois in the just 23 minutes in the opening singles.
European champions Italy face Spain in the Nations League semi-finals in Milan on Wednesday.
Jagrati Awasthi and Ankita Jain have got second and third ranks respectively.
Carlos Alcaraz is among the first four players to qualify for the Next Gen ATP Finals in Milan after his historic run to the US Open quarterfinals last week.
Daniil Medvedev lifted his fourth ATP Masters 1000 trophy in Toronto, and also earned crowns in Marseille and Mallorca.
With no players from the United States left to pull for, U.S. Open fans are adopting their neighbours from the North - Leylah and 21-year-old Felix Auger-Aliassime for the women's and men's titles.
Barbora Krejcikova beat Garbine Muguruza who won the French Open in 2016 and Wimbledon in 2017, and was the runner-up at the Australian Open last year.
BAI on Sunday named a 12-member Indian team for the Sudirman Cup, which is slated to take place from September 26 to October 3 in Finland.
Barty, who won her first Wimbledon title in July before getting upset in the first round at the Olympics, didn't drop a set in four Cincinnati matches and played only one tiebreaker.
The Indian women's team had lost to champion China in the semifinal stage in the 2016 edition of Uber Cup.
Tokyo Olympics: Bajrang Punia moves to quarterfinals after defeating Kyrgyzstan's Ernazar Akmataliev
The number two seed Indian started the match very aggressively but Kyrgyzstan's grappler defended well with his upper body strength.
The All India Tennis Association (AITA) earlier this month named a three-member Under-14 boys team comprising of Rethin Pranav RS, Kriish Ajay Tyagi and Tejas Ahuja for the tournament.
The victory at Rio de Janeiro's Maracana stadium ended Argentina's 28-year wait for a major trophy, and ended Brazil's unbeaten home record that stretched back more than 2,500 days.