Raul Jimenez put the away side in front with a half-volley from Castelo Podence's left-wing cross, before Romain Saiss's header from Neto's corner completed the scoring.
Finland's six-time champions KuPS Kuopio will face the Lithuanian champions FK Suduva in the third round of the Europa League Qualifiers. Enditem
They are currently second in Group A, behind Independiente del Valle on goal difference.
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.
Saturday's next game is a rescheduled fixture, which sees Osasuna return to their remodeled El Sadar Stadium to face a Getafe side playing their first game of the season.
New Betis coach Manuel Pellegrini made his presence felt as his side defended much better than they did last season in their 1-0 win away to Alaves last week, while Valladolid were slightly unlucky to only draw 1-1 at home with Real Sociedad.
Thomas Tuchel's side welcomed back goalkeeper Keylor Navas, defender Marquinhos and forward Mauro Icardi after the trio were among seven players tested positive for COVID-19, but missed Kylian Mbappe, who is still in self-quarantine at home.
In the opening round of CONMEBOL qualifiers, beginning on October 8, Colombia host Venezuela, Uruguay are home to Chile, Bolivia visit Brazil, Paraguay welcome Peru and Ecuador travel to Argentina. Enditem
Koeman also insisted he had a good relationship with Lionel Messi, after the striker had told the club he wanted to leave, only to later say he would stay in order to avoid lengthy legal wrangling.
His absence is likely to provide another opportunity for Cesar Dutra, who replaced Alves in the team's 2-0 defeat to Ceara in Brazil's Serie A championship on Sunday.
The next 20 minutes were dominated by Brest, but Maccabi's defense stopped all the visitors' attempts.