The SPL champions’ European adventure appeared in danger of being over before it had properly started when HJK silenced Parkhead by taking the lead two minutes after half-time through Rasmus Schuller.
However, prolific striker Gary Hooper drew the Bhoys level seven minutes later and Charlie Mulgrew’s goal just after the hour ensured Neil Lennon’s men will travel to Finland next week with a slender advantage.
Anderlecht, Maribor and Hapoel Ironi Kiryat Shmona all enjoyed convincing home wins as Helsingborgs delivered the most impressive result away from home.
Ole Gunnar Solskjaer’s hopes of setting up a reunion with his former club Manchester United in the group phase look doubtful after 10-man Moldelost 1-0 at home to Basel.
Molde’s chances of gaining a first-leg advantage were hit by Davy Angan’s red card after 27 minutes and Basel, who reached the knockout phase last term and dumped out United in the process, took advantage to score the only goal on 79 minutes through Jacques Zoua.
Anderlecht produced a dominant first-leg performance as Dieumerci Mbokani scored a goal in each half in a 5-0 mauling of FK Ekranas.
Tom de Sutter and da Rocha Kanu also scored in the first half for the Belgian side before former Liverpool player Milan Jovanovic added a late fifth goal.
Club Brugge will be the favorites to join domestic rivals Anderlecht in the next round following a 0-0 draw away to FC Copenhagen.
Fenerbahce almost suffered a surprise home defeat before salvaging a 1-1 draw at home to Vaslui following a late equalizer.
Romanian outfit Vaslui took the lead 15 minutes from time through Liviu Antal, but Bekir Irtegun spared Fener’s blushes by levelling the tie a minute from time.
Fatos Begiraj’s 14th minute goal was enough to put Dinamo Zagreb in the driving seat to progress following a 1-0 win away to Sheriff Tiraspol.
Partizan Belgrade will have to overturn a first-leg deficit to progress after Patrick Vouho’s 13th minute goal earned AEL a 1-0 home triumph.
Helsingborgs produced an impressive result on their travels after posting a 3-0 win away to Polish side Slask Wroclaw.
Alfred Finnbogason scored before half-time and second-half goals from Christoffer Andersson and Daniel Nordmark sealed the win before Slask had Rok Elsner sent off late on.
Ibrahima Sidibe turned from hero to villain as Debrecen drew 1-1 away to BATE Borisov, who reached the group stage last season.
Debrecen took the lead in Belarus through Sidibe midway through the second half, but he then scored an injury-time goal to leave the tie finely balanced.
Bryan Gerzyach scored twice in the second half to all-but send Hapeol Ironi KiryatShmona into the next stage following a thumping 4-0 home success over Neftchi Baku.
Maribor will be confident of moving on in the competition after Dejan Mezga netted in both halves during a 4-1 home win over Dudelange.
CFR Cluj hold a slender lead going into their second leg against Slovan Liberec after Ricardo Cadu’s second-half penalty pinched a 1-0 home win.
Information from The Associated Press was used in this report.