Substitute Ben Sahar headed the Israeli champions ahead in the 20th minute and the rattled Hoops were two down on 48 minutes when a mix-up between Saidy Janko and Gordon allowed Romania international Ovidiu Hoban the easiest of chances from a yard out, but the visitors held on for a 5-4 aggregate triumph.
The Scottish champions go into Thursday’s group-stage draw breathing a huge sigh of relief, knowing they will earn more than £20m and receive a incalculable boost to prestige and status.
But they made their fans suffer, including the 150 or so who made the trip to Israel.
Celtic captain Scott Brown feels they deserve their place in the Champions League group stages after a “gutsy” display in Israel.
“We managed to dig deep and it was a gutsy performance.
“It’s been a couple of years since we have been in the Champions League but we deserve our place.”
After two previous play-off defeats under previous boss Ronny Deila, the stage was set for Celtic success under the former Liverpool manager.
Rodgers brought right-back Mikael Lustig into the centre of defence at the expense of young Irishman Eoghan O’Connell, who dropped out altogether.
There was drama in the 14th minute when the Israeli side were awarded a penalty, Dutch referee Bas Nijhuis adjudging Janko to have tripped Ofir Davidzada just inside the box, but Gordon beat Maharan Radi’s blasted spot-kick away.
Hapoel soon had to make a change when former Chelsea striker Sahar replaced Lucio Maranhao, who had suffered an earlier knock, and within moments the substitute leapt to head a Radi corner past Gordon to give the hosts renewed encouragement.
Celtic were disjointed and in some disarray, and Hapoel surged forward confidently for number two.
Hapoel captain Maor Melikson went off injured to be replaced by Maor Buzaglo but it made little difference to the home side’s dominance, although Gordon was not too troubled before the interval which brought some much-needed respite for Celtic.
Rogic replaced James Forrest for the start of the second half but three minutes later there was bedlam inside the stadium when Janko appeared to knock the ball out of the hands of Gordon as he dealt with a Ben Bitton cross, leaving Hoban to tap into the empty net.
Celtic were rocking.
The traffic continued towards Gordon and he was fortunate that Sahar failed to connect properly with a volley from 10 yards.
French striker Moussa Dembele came on for the dejected Griffiths and moments later shaved the far post with a curling shot on a Celtic counter-attack.
In the 63rd minute Sinclair got the break of the ball six yards out but Goresh made a fine save before Dembele’s shot on the follow-up was blocked.
But at the other end Hoban’s spectacular scissors-kick escaped the far post before he had a long-range effort saved by the Scotland goalkeeper.
The home side probed down the right and left but ran out of ideas and then time - four minutes of which were added - leaving Celtic shocked but through to the first round proper in the Champions League.
Goresh, Bitton, Vitor (Shabtai 75), Tzedek, Davidadze, Radi, Ogu, Hoban, Melikson (Buzaglo 34), Nwakaeme, Maranhao (Sahar 19).
Gordon, Lustig, Toure, Tierney, Janko, Bitton, Brown, McGregor (Sviatchenko 65), Sinclair,Forrest (Rogic 46), Griffiths (Dembele 57).
Bas Nijhuis (Holland).