Celtic was drawn at home first against Stjarnan in the second qualifying round, and Steaua will travel first to Trencin.
Malmo was paired with Zalgiris of Lithuania and Ludogorets Razgrad is at home against Moldova's champion Milsami Orhei.
Another Champions League regular, Dinamo Zagreb, is away first against Fola Esch of Luxembourg.
The 17 second-round pairings drawn Monday involve champions of Europe's lower-ranked national leagues.
First-leg matches are played July 14-15 and second legs on July 21-22.
UEFA also drew four first-round qualifying pairings which are played between June 30 and July 8.
High-placed teams from Europe's strongest leagues, including Manchester United, enter at the playoff round in August.