The Bhoys knew they needed to overcome Hapoel at Parkhead and hope that Hamburg did not get the better of Rapid Vienna in their meeting in Germany.
But despite a much-improved European performance from the Scottish Premier League side in which they kept up their end of the bargain, Hamburg's own 2-0 victory confirmed their passage to the last 32 alongside Hapoel.
Two defeats and two draws from their four opening games had led to open season on Celtic from the critics.
However, the Bhoys refused to go down without a fight and after a patchy start from both sides, they had the ball in the net after four minutes but Scott McDonald's effort was correctly ruled out for offside.
Visiting keeper Vincent Enyeama was called into action in the 20th minute when he palmed away McDonald's shot after a strong run from the lively Aiden McGeady.
Itay Shechter sounded a warning when he crashed a shot just over the crossbar from 12 yards after a quick free-kick move threatened to open up the home defence.
But Celtic broke the deadlock moments later when McGeady picked out the run of the late arriving Georgios Samaras and he found the back of the net with a powerful downward header from six yards.
Barry Robson came close to doubling the advantage when he curled a free-kick just over the bar from 25 yards.
However, he should have done better on the stroke of half-time when he crashed a close-range shot straight at the body of the diving keeper when following up a Samaras shot which cannoned back off the upright.
Shechter brought a first save of the game out of Lukasz Zaluska on the hour mark with a shot from distance as the visitors upped their game following a change in personnel.
But Robson was not to be denied and the diminutive midfielder did finally double the advantage on 68 minutes when his 25-yard free-kick found the bottom corner of the net after substitute Marc-Antoine Fortune was felled needlessly.
Hapoel continued to huff and puff in the closing minutes but Celtic closed out the game with few problems.
Fortune slipped as he prepared to make it 3-0 in the 83rd minute and McDonald wasted a gilt-edged chances soon after, but it mattered little as Celtic's European fate was sealed elsewhere on a night of mixed emotions for the Parkhead faithful.