Australia midfielder Tom Rogic was a late substitute as Celtic came from behind to claim a 4-1 win over Stjarnan and seal a 6-1 aggregate success in the UEFA Champions League (UCL) second-qualifying round.
Rogic has made an impressive return this season from a series of chronic injuries which hampered his first two years at the Scottish giant, with a number of standout performances for the club's development squad.
He was given a late cameo in the 4-1 win over Stjarnan, replacing Nadir Ciftci in the 82nd minute.
The Hoops were 2-1 up when the Socceroos midfielder joined the match, with two late goals sealing a comprehensive win.
Leading 2-0 from the first leg at Parkhead, Celtic was always favourite to progress as the expense of the Icelandic champion.
But it looked as if the Scottish Premiership side was going to be in for a tough examination when Stjarnan took an early lead through Olafur Karl Finsen's low strike.
Forward Finsen capped off a free-flowing attack, but Celtic dominated proceedings thereafter and eventually levelled as its extra quality told.
Midfielder Nir Biton headed home from a corner before Charlie Mulgrew put Celtic ahead in the 49th minute, with Leigh Griffiths and Stefan Johansen rounding off the scoring late on.
Mulgrew collected the ball from Scott Brown and unleashed a shot past goalkeeper Gunnar Nielsen, ending the tie as a contest and setting up a third-qualifying round meeting with Qarabag of Azerbaijan.
Stjarnan opened the scoring after just seven minutes as Finsen finished off a wonderfully-worked team move.
Celtic still enjoyed the lion's share of possession and looked to have equalised in the 22nd minute through Dedryck Boyata, who bundled in on the rebound from a Johansen shot only to be flagged for offside.
But Ronny Deila's men did restore parity 11 minutes later when Biton capitalised on some poor defending to glance Johansen's right-wing corner into the net at the near post.
And Celtic almost completed the turnaround in the 36th minute, Stuart Armstrong shooting narrowly wide of the post from Gary Mackay-Steven's pass.
Just four minutes into the second half, though, Celtic did take the lead on the night and took an even firmer grip on the tie thanks to Mulgrew.
The Scotland international picked the ball up on the edge of the area and arrowed a shot into the bottom-right corner beyond a helpless Nielsen.
There were chances at both ends as the game opened up in the second half. Substitute Griffiths shot just beyond the far post for Celtic before Brynjar Gudjonsson and Finsen went close for the host.
Mackay-Steven and Nadir Ciftci spurned further opportunities for Celtic before Griffiths rolled the ball into an empty net and Johansen fired home a fourth to give the scoreline added gloss.
Steaua Bucharest advanced despite a 3-2 defeat to Trencin.
The Romanian side, which won the European Cup in 1986, led 2-0 from the first leg in Slovakia.
However, Trencin levelled the tie thanks to goals from Wesley and Matus Bero, only for Steaua to reassume command as Sulley Muniru and Gregory Tade hit the target, the latter doing so from the penalty spot.
Despite Milan Rundic seeing red for a second bookable offence, Wesley netted a second to put Trencin within a goal of qualification, but Steaua hung on to set up a tie with Partizan.
Dinamo Zagreb came through in more comfortable circumstances, a double from Marko Pjaca helping the Croatian champion to a 3-0 win over Fola Esch, ensuring a 4-1 aggregate triumph.
Qarabag claimed a 1-0 victory at Rudar Pljevlja to go through.
The New Saints suffered extra-time heartbreak at Videoton, losing 2-1 on aggregate against the Hungarian side.
Matthew Williams levelled the tie and forced extra-time by netting for the visitor in the 78th minute, but a 107th-minute effort from Videoton midfielder Adam Gyurcso proved decisive, with Sam Finley seeing red for The New Saints 11 minutes later.
Lech Poznan beat Sarajevo 1-0 to progress, BATE progressed after a 0-0 draw at Dundalk and Astana turned around a 1-0 first-leg deficit to knock out Maribor with a 3-1 home win in Kazakhstan.