Olympiakos scored three late goals to beat Russian debutant Krasnodar 4-0, and Dinamo struck twice in the first half hour in a 2-0 win over Rosenborg.
Young Boys was the only home team not to win, needing a 76th-minute penalty awarded by video review — and scored by French veteran Guillaume Hoarau — to secure a 2-2 draw with Red Star Belgrade.
Red Star, playing its seventh qualifying game in seven weeks, is trying to advance from the first qualifying round to the group stage in back-to-back seasons.
Olympiakos played the first of its 18 Champions League campaigns more than a decade before Krasnodar was founded in 2008.
The Greek runner-up took the lead in the 30th through Spanish forward Miguel Ángel Guerrero. A flurry of goals from the 78th minute by Lazar Randelovic, twice, and Daniel Podence gave Krasnodar a daunting task in the second leg next Tuesday.
Once a Champions League regular, Rosenborg trailed to an early penalty by Dinamo's Bruno Petkovic, and Mislav Orsic doubled the lead in the 28th.
Red Star, the 1991 European champion, trailed to a seventh-minute goal by Young Boys forward Roger Assalé that was leveled in the 18th by Australia defender Milos Degenek's looping header.
Mateo Garcia put the Serbian champion ahead a minute into the second half, before Hourau came off the bench to level from the spot. The 35-year-old former France international went close to winning the game for the Swiss champion with a glancing header in the 89th.