The Clarets head to Turkey on Thursday for the first leg before they attempt to seal progression on home soil seven days later.
If they can follow-up their hard-fought triumph against Aberdeen with another memorable victory, just two winnable games will separate them from the main draw.
Greeks with European pedigree
Olympiacos, who finished 3rd in the Greek league last season, will be the favourites in their meeting with Luzern.
They have European pedigree, having recorded a number of notable triumphs in the Champions League.
In 2013/14, they beat Manchester United 2-0, and the following year they got the better of both Atletico Madrid and Juventus.
Most recently, they recorded a stunning 3-2 victory against Arsenal at the Emirates.
Their 2017/18 European campaign was less successful as they finished bottom of their Champions League group.
They had to face Barcelona, Juventus and Sporting Lisbon, but they would still have been disappointed with their return of only a single point.
On the whole, it was a tricky campaign which saw their run of seven consecutive domestic championships come to an end.
Possible trip to Switzerland
Luzern, meanwhile, finished third in last season's Swiss Super League but have started this season slowly.
They currently sit in eighth place after losing two of their first three games.
Rene Weiler's side also earned a place in the qualifiers last year but fell to Croatian team Osijek in the second round.
The playoff ties will be played on the 24th and 31st of August.
Burnley may be forced to rest key players for their opening Premier League games against Southampton, Watford, Fulham and Manchester United, all of which could directly follow a European game.