Dejan Mezga and Brazilian Marcos Tavares scored from free kicks before half-time in the third qualifying round, first leg tie and Mezga was on target again two minutes after the re-start for the Slovenian double winners.
Robert Beric made it 4-0 in the 77th minute before Aurelien Joachim pulled one back with his seventh goal in five games in the competition for Dudelange, who overcame Salzburg on away goals after a 4-4 aggregate draw in the last round.
Hapoel Kiryat Shmona, a small-town team who won their first Israeli championship last season, thumped Azerbaijan champions Neftci Baku 4-0 as they continued their surprise progress.
Kiryat Shmona, Israel's northern-most urban centre with a population of 23,000, was for decades the target of rockets and cross-border raids mainly by Palestinian militants based across the border in Lebanon.
Hapoel played the match in Haifa as their own stadium does not meet Champions League requirements.
Belgian champions Anderlecht, who last qualified for the group stage six years ago, hammered Lithuanian champions Ekranas Panevezys 5-0 and former European champions Celtic overcame HJK Helsinki 2-1 at home.
Cameroon forward Jacques Zoua scored a late winner for Swiss champions FC Basel in a 1-0 win away to Ole Gunnar Solskjaer's Molde in Norway.
Debrecen's Ibrahim Sidibe had an eventful match at BATE Borisov, coming on as a substitute for the last half hour to score both goals in a 1-1 draw.
The Senegalese striker met Adamo Coulibaly's to score with his first touch in the 67th for the Hungarian champions, only to inadvertently turn Aleksandr Pavlov's cross into his own net in stoppage time to give the Belarus champions a draw.