The Scottish champions held a slender one-goal lead heading into the second leg at Murrayfield, where they were playing due to Parkhead being used for the Commonwealth Games, and they made light work of moving one step closer to the group stages on Tuesday.
Defender Van Dijk was on hand to open the scoring from close range after goalkeeper Stefan Logi Magnusson failed to deal with a Kris Commons corner 13 minutes in, then doubled his side's lead seven minutes later following another error from the keeper and poor defending.
Icelandic outfit KR fell further behind when Pukki fired home after good work from Callum McGregor 27 minutes in and the Finland striker added a fourth 19 minutes from time to complete the rout, with Commons denied a fifth goal by the post.
Celtic now face either St Patrick's Athletic or Legia Warsaw, who meet in Dublin on Wednesday, in the third qualifying round.
Belarusian champions BATE Borisov scraped through on away goals as they held Skenderbeu to a 1-1 draw in Albania.
After a tense goalless first leg it was BATE who edged in front when midfielder Edgar Olekhnovich opened the scoring with 29 minutes gone, but the home side responded well and Gerhard Progni was denied by the woodwork three minutes later.
Skenderbeu did get their equaliser courtesy of Croatian defender Marko Radas after 67 minutes, then substitute Andi Ribaj struck the frame of the goal two minutes from time in a tense finale as BATE held on.
Sparta Prague coach Vitezslav Lavicka had the luxury of making eight changes to the side that hammered Levadia 7-0 in the first leg last week as they drew 1-1 to progress 8-1 on aggregate.
Captain David Lafata was given a rest after scoring five goals in the first leg and Ingemar Teever struck for the home side in Estonia after Lukas Marecek had put Sparta in front and it finished all square on the night.
Qarabag made light work of ending Valletta's participation in the competition as they secured a resounding 4-0 win, having earned a 1-0 victory in the first leg, as the visitors had Jonathan Caruana and Ryan Fenech sent off.
Sheriff coasted through 5-0 on aggregate by beating Sutjeska 3-0 away from home in the second leg, while Ludogorets also progressed after drawing 1-1 at Dudelange - having won the first leg 4-0.
Maccabi Tel Aviv beat Andorran outfit Santa Coloma 2-0 to go through 3-0 on aggregate, but Barak Badash was sent off late on after coming off the bench before Albert Mercade saw red for the visitors.
Marko Milinkovic struck twice in the last 16 minutes as Slovan Bratislava won 2-0 to seal their passage into the third qualifying round at the expense of The New Saints, who they beat 1-0 in the first leg.
Dinamo Zagreb and Partisan Belgrade also made it through by beating Zalgiris and HB 6-0 and 4-1 on aggregate respectively.
