The offside flag deprived the Hoops of two goals in the first 12 minutes following the loss of Jozo Simunovic to injury before Mohamed Elyounoussi stunned Parkhead with a 20-yard drive.
Midfielder Kris Commons levelled five minutes later but Molde's 40-year-old skipper Daniel Hestad restored the Norwegian club's lead in the 37th minute to give them the interval advantage.
The hapless home side's misery was compounded in the 77th minute when Nir Bitton, who had moved from midfield to defence in a reshuffle, was shown a straight red card by referee Slavko Vincic for a 'last man' tackle on Molde substitute Ola Kamara.
Defeat kept Deila's side bottom of Group A with two points from four games and w ith a home game against Ajax and a trip to Fenerbahce remaining, the Scottish champions require two wins to progress, an unlikely scenario on the evidence so far
It was a huge night for Deila and ostensibly, it looked to have started in positive fashion when Croatian defender Simunovic shrugged off a foot knock to take his place in defence.
However, the 21-year-old lasted only six minutes before he was left on the turf following a challenge from Molde attacker Tommy Hoiland, and hobbled off to be replaced by Tyler Blackett.
Even then it did not look as if it was going to be Celtic's night.
In the 10th minute Bitton drove in a pass from Commons but the offside flag was up for Leigh Griffiths, who had shifted to get out of the way.
Two minutes later, Griffiths thought he had scored from close range with a shot but again he was ruled offside to further anger the Hoops support.
An anxiety enveloped Parkhead every time Molde broke forward and in the 19th minute, Hoiland raced clean through on goal but his poor touch allowed Craig Gordon to gather.
However, the Hoops goalkeeper had no chance in the 21st minute when Blackett's header was returned from 20 yards by Elyounoussi into the far corner of the net.
The visitors, though, looked far from cohesive in defence and Commons lifted the despondency when he wrong-footed Molde keeper Ethan Horvath by getting a touch on Bitton's 25-yard drive, with the ball speeding in at the post.
The home side almost took the lead in the 31st minute after a mistake by Horvath but the keeper saved Tom Rogic's shot before Commons' drive was headed behind for a corner by Hestad.
The veteran midfielder's next contribution was even more meaningful when, after getting in front of Blackett at the far post, he turned Elyounoussi's deflected cross into the net after the home side failed to properly clear a corner.
James Forrest replaced Commons for the start of the second half with Kamara on for Hoiland and both teams continued in the same, almost reckless, fashion.
Amid some smart attacking play by Molde, Horvath tipped a powerful drive from Celtic left-back Kieran Tierney over the bar for a corner, which came to nothing, and then pulled off a brilliant save down at his post from a Mikael Lustig header, before denying Bitton with another great block.
Twice within minutes, however, Hoops defender Dedryck Boyata had to clear from in front of his own posts.
In the 69th minute, substitute Blackett was replaced by striker Nadir Ciftci with Bitton dropping back to defence, before Griffiths blazed the ball wide from 10 yards after being picked out by Stefan Johansen.
The Scottish champions kept plugging away, all the while leaving themselves vulnerable at the back and when Bitton was sent packing for fouling Kamara as he raced through on goal, it capped a disastrous night for the Hoops.
Celtic were consequently and understandably deflated but it could have been worse - with seven minutes remaining Kamara flashed a drive from distance off the top of the crossbar - before the visitors managed four added minutes with some comfort.
TWEET OF THE MATCH
"Celtic need a miracle now!" - Former Celtic forward John Hartson (@JohnHartson10) sums up the Scottish side's chances of advancing to the knockout stages of the Europa League.
PLAYER RATINGS
Celtic
CRAIG GORDON 7 (out of 10)
MIKAEL LUSTIG 6
DEDRYCK BOYATA 5
JOZO SIMUNOVIC: 5
KIERAN TIERNEY 7
NIR BITTON 7
STEFAN JOHANSEN 5
STUART ARMSTRONG 6
TOM ROGIC 7
KRIS COMMONS 6
LEIGH GRIFFITHS 6
Substitutes:
TYLER BLACKETT: Replaced the injured Simunovic in the ninth minute then was subbed for Nadir Ciftci
5.
NADIR CIFTCI: Replaced substitute Tyler Blackett
5
STAR PLAYER
Celtic's 18-year-old left-back Kieran Tierney was the best of the home side by some distance with some fine attacking play and he also came close to scoring.
MOMENT OF THE MATCH
When Jozo Simunovic hobbled off with an injury after six minutes
From then on, it was a struggle for the home side.
VIEW FROM THE BENCH
After two points from three Group A matches Hoops boss Ronny Deila needed the win to get Europa League hopes back on track - and failed
He will come under pressure while Ole Gunnar Solskjaer will be delighted with the way his side performed.
MOAN OF THE MATCH
Exciting as it was on occasions, the standard of play for a European match was at times deplorable, with mistakes and unforced errors from both sides thoughout the 94 minutes.
WHO'S UP NEXT?
Ross County v Celtic (Ladbrokes Premiership, Sunday 8 November).
Source : PA
Source: PA