Four-time European champions Ajax laboured to a 1-1 draw at home to PAOK in the first leg of Tuesday's Champions League third qualifying round, while Shakhtar Donetsk overcame Young Boys 2-0.
Djalma Campos handed PAOK a first-half lead in Amsterdam, and although Kasper Dolberg equalised before the hour Ajax still face a tricky task in next week's second leg in Greece.
In Lviv, Shakhtar Donetsk defeated Switzerland's Young Boys with goals in either half from Brazil international Bernard and Ukraine striker Yevhen Seleznyov as the visitors saw Milan Vilotic sent off.
Belarus outfit BATE Borisov, who are seeking to reach the group stage for the fifth time in six seasons, beat Irish side Dundalk 1-0, while Dinamo Zagreb eased past Georgia's Dinamo Tbilisi 2-0. Rostov, last year's surprise Russian Premier League runners-up, twice fought back from a goal down to earn a 2-2 draw at home to Anderlecht in their Champions League debut.
On Wednesday, Brendan Rodgers' Celtic make the long trip to Kazakhstan to face Astana, who made their group stage debut in last year's competition. Fenerbahçe host 2004 finalists Monaco in an intriguing third-round tie, while Olympiacos are at home to Israel’s Hapoel Beer-Sheva.