The Scottish champions dominated the ball in the one-legged play-off tie in Bosnia and Herzegovina without really testing home goalkeeper Vladan Kovacevic in the first half.
Celtic continued in relentless pursuit of a goal and in the 69th minute the French striker, in for injured Albian Ajeti, knocked the ball in from close
after Kovacevic failed to deal with a drive from the excellent Ryan Christie.
The Parkhead side will deservedly take their place in group stage draw on Friday.
Neil Lennon was so pleased with his side's 3-0 defeat of Hibernian at the weekend he made only one change, enforced, with Edouard replacing Ajeti.
However, there was another injury blow for the Hoops boss after only nine minutes when defender Nir Bitton failed to shrug off the effects of a foul by Aleksandar Pejovic and was replaced by Israel team-mate Hatem Elhamed.
Celtic kept stride and moved the ball around the pitch well for long spells but in the 13th minute, in one of Sarajevo's rare breaks, Parkhead keeper Vasilis Barkas pushed an angled drive from Benjamin Tatar around his near post, the corner coming to nothing.
The home side, much changed from the one that lost home and away to Celtic in the first Champions League qualifier last season, were well organised defensively.
Celtic did create an opportunity five minutes before the break when Edouard set up Mohamed Elyounoussi but his shot from 16 yards was comfortably saved by Kovacevic.
Seconds from the interval, as Sarajevo countered with pace again, former West Ham winger Matthias Fanimo had a fresh air swipe on the angle from 12 yards.
Sarajevo continued to frustrate after the break as Celtic kept pressing.
In the 52nd minute Celtic defender Kristoffer Ajer stabbed a Christie corner past from six yards and then Elyounoussi's drive from 20 yards was saved by Kovacevic
A long-range effort from Christie in the 63rd minute just cleared the bar and Elhamed volleyed wide from a Christie corner.
Sarajevo could hardly get out of their own half although earlier defender Hrvoje Milicevic had headed a corner past the far post
However, eventually the breakthrough came with Edouard pouncing when Kovacevic fumbled a drive from Christie to steer the ball over the line from eight yards.
Sarajevo's attitude changed but Christie fancied his chances and stung the palms of Kovacevic with another powerful drive.
There was something of a late rally by Sarajevo but nothing Celtic could not handle and they can now look forward to European football for at least the rest of the year.