The Scottish champions were stunned in Bosnia and Herzegovina when Mirko Oremus opened the scoring for the home side in the 29th minute following a corner.
However, Neil Lennon's side were soon back level as 20-year-old Mikey Johnston thundered in a leveller from 25 yards six minutes later.
French striker Odsonne Edouard started his season with a classy finish in 51st minute before substitute Scott Sinclair, on for James Forrest, added a third with a cheeky finish.
It was the first qualifying game of a potential four rounds to make the group stages and the Glasgow club clearly have the upper hand.
The return game is at Parkhead next Wednesday, with the victors taking on the winners of the tie between Estonian side Nomme Kalju and Shkendija of North Macedonia.
On a sodden night in Sarajevo, Belgian left-back Boli Bolingoli, signed from Rapid Vienna for a reported fee of £3 million, made his debut and there was a place for midfielder Ryan Christie, back from his face injury which kept him out since April.
Striker Edouard led the line while fellow Frenchman Christopher Jullien, signed from Toulouse for a reported fee of £7 million, started on the bench.
It was a struggle for both sets of players to adapt to the slippery conditions but it was the home side who had the first chance after seven minutes when attacker Anel Hebibovic outstripped the Hoops defence only for keeper Scott Bain to block with Celtic eventually clearing their lines.
The rain continued to cascade and Celtic were undone when another Slobodan Milanovic corner broke loose in the box and Oremus reacted quickly to slide it in from a few yards out.
However, Celtic showed their mettle. Christie had a decent free-kick saved by Vladan Kovacevic but the Sarajevo keeper had no chance when Johnston got the ball out from under his feet and hammered a drive high into the net.
The Parkhead side could have taken the lead before the break when Edouard played in Forrest but his effort with the outside of his foot flew over the bar.
Six minutes into the second half, it looked even better for Celtic when Edouard took a pass from Forrest and calmly moved into the box before lifting the ball over Kovacevic, leaving two defenders trailing.
The goal eased the nerves of the 300 or so travelling fans and Sinclair made entry into the next round a formality when he back-heeled home a cross from fellow substitute Lewis Morgan from six yards out.