Bertrand Traore scored for Ajax in their 2-1 win at nine-man Panathinaikos in a game that saw three red cards.
Baba Rahman grabbed the only goal in Schalke's victory at Nice, for whom ex-Liverpool striker Mario Balotelli failed to add to his two debut goals.
Frank de Boer's Inter Milan - in Southampton's Group K - lost 2-0 at home to Hapoel Be'er Sheva.
Inter, who have won the European Cup and Uefa Cup three times each, have picked up four points from their opening three games in Serie A since De Boer replaced Roberto Mancini.
Hapoel Be'er Sheva were knocked out of the Champions League by Celtic at the play-off stage earlier this season.
Zenit St Petersburg pulled off one of the Europa League's great comebacks when they came back from 3-0 down at Maccabi Tel-Aviv to win 4-3, with three of their goals coming in the last six minutes. They are in Group D with Dundalk.
Jan Sykora scored the fastest goal in Europa League history - after 10.69 seconds - for Slovan Liberec at Qarabag. They ended up drawing 2-2, with Milan Baros, who played for Liverpool, Aston Villa and Portsmouth, scoring their other goal.
In Manchester United's Group A, Simon Kjaer scored in the sixth minute of stoppage time for the visitors as Fenerbahce drew 1-1 at Zorya Luhansk.
Alexandre Pato, who had a loan spell at Chelsea last season, scored in Villarreal's 2-1 win over FC Zurich. Facundo Ferreyra, who never played a single minute for Newcastle in a disastrous 2014-15 loan spell, scored Shakhtar Donetsk's goal in a 1-0 win at Konyaspor.
Former Manchester United striker Giuseppe Rossi netted for Celta Vigo in their 1-1 draw at Standard Liege.
Italian minnows Sassuolo, in their first European campaign, thrashed Athletic Bilbao 3-0, while Roma drew 1-1 at Viktoria Plzen.
Two Chelsea loanees were among the scorers in the opening fixtures of the Europa League group stage.