Neil Lennon's side had put themselves in a strong position with a first-leg win and were rarely tested at Celtic Park, triumphing 2-0 on the night thanks to strikes from Efe Ambrose and Georgios Samaras.
The Nigerian defender met a cross from Anthony Stokes to head past Conor Devlin in the 16th minute. Kris Commons and James Forrest both had good chances to extend Celtic's lead but it was Samaras who added the second goal with a header from Commons' cross late in the game.
BATE Borisov's defeat to Shakhter Karagandy was the shock of the round, the Kazakh side winning 1-0 on the night to progress 2-0 on aggregate.
The Belarusian side, who beat eventual champions Bayern Munich in the group stages of the competition last season, failed to score against their Kazakh opponents in the 180 minutes of football.
After losing the first leg 1-0, they succumbed to the same result in Karagandy, as Igor Zenkovich scored the only goal of the game with eight minutes to play.
The result puts Shakhter into the third round of qualifying for the first time in their history.
After their 7-1 victory in the first leg, Swedish side Elfsborg cruised to a 4-0 win against Daugava Daugavpils in Latvia, taking them to an emphatic 11-1 aggregate triumph.
James Keene scored the only goal of the first half, walking the ball into the back of the net just before half-time, while Viktor Claesson , Niklas Hult and a penalty from Martin Andersson sealed the win for the Swedish champions.
After a goalless first leg, Estonian champions Kalju overcame HJK Helsinki 2-1 in Tallinn, goals from Yankuba Ceesay and Damiano Quintieri put the hosts ahead and although Demba Savage got a goal back in the second half, Kalju were able to hang on.
Moldovan side Sheriff produced an emphatic away performance in Montenegro to beat Sutjeska 5-0 and record a 6-1 aggregate victory, while Molde ended Sligo Rovers' campaign with a 2-0 victory which saw them progress 3-0 on aggregate.
Dinamo Tbilisi were in complete control against Faroe Islands' EB/Streymur, adding three more goals to the six they scored in the first leg to win 9-2 on aggregate. Goals from Xisco, Jaba Dvali and Darko Glisic saw them win 3-1 on the night, with Niels Pauli Danielsen getting a consolation strike for the hosts.
Maccabi Tel Aviv built on their first-leg victory over Hungarian side Gyor to progress 4-1 on aggregate, the Israelis winning 2-1 to book their place in the third round of qualifying.
Former European Cup winners Steaua Bucharest had on-loan striker Federico Piovaccari and Alexandru Bourceanu to thank for firing them into the next round, the Romanian side winning 2-1 against Vardar to seal a 5-1 win over the Macedonians.
Nigerian Nurudeen Orelesi was Skenderbeu's hero as they beat Neftchi 1-0 after extra time. Neither side were able to break the deadlock in the first 180 minutes of the tie but Orelesi struck with four minutes to play in the added period to seal the Albanian side's progression from round two for the very first time.
Dinamo Zagreb were never in danger of not making the third round after their comfortable 5-0 first-leg win over Luxembourg's Fola, and the Croatian side came through the second leg, winning 1-0 thanks to striker Andrej Kramaric's 78th-minute penalty.
Viktoria Plzen and Zeljeznicar shared 10 goals in their tie, the Czech side coming out 6-4 victors on aggregate after a 2-1 second-leg victory in Sarajevo.
Meanwhile, Bjorn Daniel Sverrisson and Atli Bjornsson were FH's heroes as they came from a goal behind to beat Ekranas 2-1 on the night, sending the Icelandic champions through 3-1 on aggregate.
