On a night of high drama in Poland, the visitors took the lead through Callum McGregor in the seventh minute but proceeded to collapse, with Legia striker Miroslav Radovic scoring a brace before the break and defender Efe Ambrose was red-carded a minute from the interval for a professional foul on Michal Kucharczyk.
In the second half, 10-man Celtic were let off the hook when Legia captain Ivica Vrdoljak missed the target with a penalty, after Charlie Mulgrew was adjudged to have fouled Ondrej Duda inside the box, but in the 84th minute Michal Zyro headed in number three.
Mulgrew conceded another penalty with a trip on substitute Jakub Kosecki, though Fraser Forster saved as Vrdoljak failed with a second spot-kick, but a fourth was scored when Kosecki slotted in a pass from Lukasz Broz
Unless the Hoops can overturn the three-goal deficit next week at Murrayfield in Edinburgh - used as Celtic Park is still out of commission due to its use in the Commonwealth Games - Deila's men face the dispiriting prospect of dropping into the Europa League
Afterwards the Norwegian said: "It was a very poor performance and worst of all was the commitment, the energy, I need much more energy in the game.
"We didn't stay together for 90 minutes.
"We started well and got the first goal but then gave way to pressure and we lost our heads and started giving the ball way in midfield so many times and mistakes in the back four - some were high, some were low, instead of being together.
"But what is done is done, we have to learn from this very quick, we got the away goal and we still have a chance.
"It is possible to win 3-0 again - like we did last year in the play-off - with the help of the supporters and we will do everything we can to do that.
"But we have to learn from this week
We need a whole different attitude and discipline in our play."
After watching his side turn in a disjointed performance, Deila vowed to make himself much clearer in his instructions ahead of the second leg.
He said: "It was not the plan to go forward at 2-1
We had a spell when we had good control but started making stupid mistakes and they got space behind us.
"That is my responsibility
I have to be much clearer as to what I expect and how we perform as a team, defensively and offensively, and tonight we lost our heads and that is not good enough."
Deila was not too harsh on Ambrose, whose defending was ropey before he was sent packing.
He said: "It is not only that situation, it was a consequence of many situations.
"We lost the ball too much in middle, the back four was not together and when we lost the ball we were too open.
"We made it difficult for ourselves and in the end they got a counter-attack and Efe went in, but it was more of a consequence of bad defending."
Legia boss Henning Berg, who used to play for Celtic's traditional rivals Rangers, was pleased by the way his side kept going despite the disappointment of missing two penalties.
He said: "When we missed the two penalties we didn't feel sorry for ourselves, we pushed and got more goals and we could have scored more.
"It is not normal to miss two penalties, especially for Ivica as he has never missed for me.
"Ivo was confident but that is football
But I am very happy with the way we played and so happy for my players to show what we can do in a game like this."
Source : PA
Source: PA