Odsonne Edouard rewarded Celtic for a much-improved second-half performance in Sarajevo as his winner secured their place in the Europa League group stage.
The French striker poked home 20 minutes from time to settle the play-off after goalkeeper Vladan Kovacevic spilled a Ryan Christie shot.
Neil Lennon's side were rarely troubled as they progressed past the qualifiers for the third year in succession.
Celtic are in pot one for Friday's draw in Nyon (12:00 BST).
Having had their Champions League hopes extinguished by Ferencvaros, Lennon's men capitalised on their second chance in Europe.
But they were made to work for it in Bosnia-Herzegovina after a sluggish first 45 minutes. It took until shortly before the interval to register a shot on target, Mohamed Elyounoussi's volley from an Edouard lay-off proving an easy gather for the keeper.
Celtic were given an early warning at the other end when the hosts broke at pace and Benjamin Tatar lashed a near-post drive into the side-netting.
A big improvement was required from Lennon's men, and it was duly delivered. Kristoffer Ajer scuffed wide from six yards after a corner dropped at his feet, then Elyounoussi hammered a shot at the keeper.
The second-half momentum kept building, with Christie thumping a shot from distance inches over and substitute Hatem Elhamed volleying narrowly wide.
The nerves were finally eased when Edouard - restored to the starting line-up in place of the injured Albian Ajeti - followed up on Christie's shot to score his sixth goal of the season.
Christie and Jeremie Frimpong had chances to make the closing stages more comfortable, but Celtic saw out the game with minimum fuss.
Man of the match - Ryan Christie
What did we learn?
Celtic were on a hiding to nothing in these qualifiers, but can breathe easy now and look forward to challenging themselves against elite opposition. The finance from another group-stage campaign will come in handy too.
And although Edouard has yet to hit top form this season, he remains integral to Celtic's hopes of success. The 22-year-old once again proved he is a player for the big occasion, pouncing to breach the stubborn hosts who had repelled heavy pressure.
What did they say?
Celtic manager Neil Lennon: "It was a dominant performance against a strong, physical and well organised team.
"The pitch was awful. We had to adapt and be more aggressive in our play, put more crosses in and try to switch the play quicker. We did that really well second half."
Celtic sign off before the international break with a trip to St Johnstone (12:00 BST) in the Scottish Premiership on Sunday.
- 6OremusBooked at 90mins
- 29DupovacBooked at 43mins
- 24DjokanovicSubstituted forJukicat 85'minutes
- 38Olubori Adebowale FanimoSubstituted forVelkoskiat 85'minutes
- 9AhmetovicSubstituted forHandzicat 68'minutes
- 6BittonSubstituted forElhamedat 10'minutes
- 30FrimpongBooked at 89mins
- 17ChristieSubstituted forNtchamat 86'minutes
- 22EdouardSubstituted forKlimalaat 78'minutesBooked at 90mins
- Benoît Bastien
- Shots on Target