The hero of the night became the home team's Moldavian striker Ion Nicolaescu who tied the game five minutes before the end of regulation on his debut and then put Dunajska Streda ahead 4-3 six minutes into overtime with a quick and un-saveable shot from the edge of the box.
Nishikori needed six match points to seal the win in two hours at the ATP Masters event held behind closed doors at the Foro Italico.
Despite Thursday's result, Flamengo are second in their Libertadores group with six points from three matches but with a negative goal difference. The Rubronegro remained in Ecuador after the clash in Quito to prepare for Tuesday's fixture against Barcelona SC in the Pacific port city of Guayaquil.
"I even couldn't believe that a defender has terrific skills like Li. It's important to have such an advantage in the game," Suning coach Cosmin Olaroiu praised.
"He has the speed, he can combine, can defend, can press. It makes it just more unpredictable and gives us real options for different systems because he can play pretty much all three positions up front in a 4-3-3. If we play with four midfielders he can play both wings," said the German coach.
After a goalless 90 minutes, the visitors were close to victory in extra-time when their Slovenian striker Zan Medved scored with a beautiful header after a long center-pass with only nine minutes left.
Battaglia made his debut for Argentina against Colombia in September 2018 but his ascent on the international stage was halted by a serious knee injury just weeks later.
Along with fellow new boy Allan, the Colombian was lively throughout and the Toffees fully deserved their win, with Dominic Calvert-Lewin's thumping header proving decisive.
Hoffenheim dominated possession and produced more promising chances but Horn was on guard to deny Kramaric and Pavel Kaderabek in the 18th minute.
After selling Ivan Rakitic to Sevilla, Barca are currently looking to sell striker Luis Suarez, with Juventus a possible destination, while Arturo Vidal is close to joining Inter Milan.
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