Callum McGregor gave the visitors the lead in the seventh minute with a strike from outside the penalty area, the 21-year-old midfielder's second European goal in three games, and that looked like being the early boost that would propel the Scottish champions on.
However, Legia striker Miroslav Radovic levelled three minutes later and then notched another before the break, with Parkhead defender Efe Ambrose red-carded a minute from the interval for a last-man challenge 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 captain Charlie Mulgrew was adjudged to have fouled Ondrej Duda inside the box.
But in the 84th minute Michal Zyro headed in number three before Mulgrew conceded another penalty with a trip on substitute Jakub Kosecki, this time Fraser Forster saving Vrdoljak's spot-kick.
But there was another sting in the last seconds when Kosecki slotted in a pass from Lukasz Broz to stretch Legia's lead ahead of next week's return game at Murrayfield in Edinburgh - used as Celtic Park is still out of commission due to its use in the Commonwealth Games.
It was quite a night.
Manager Ronny Deila gave new signing Jo Inge Berget his Celtic debut despite the forward only arriving from Cardiff on a six-month loan deal on Monday - but it was a product of Celtic's youth academy who made the first impact.
McGregor, who had scored the winner against KR Reykjavik on his European debut in the previous round, got the Scottish champions off to a great start when he drifted in off the right flank past left-back Tomasz Jodlowiec before firing in a left-footed drive from 20 yards which gave home keeper Dusan Kuciak no chance.
However, barely had the cheers of the 160 or so Celtic fans inside the 31,000-capacity stadium subsided when Legia drew level as the Parkhead defence dropped their guard.
When Legia's highly coveted midfielder Zyro played in Radovic the Croat had plenty of time inside the Celtic box to get his shot away, with Hoops keeper Forster getting a touch but unable to keep it out.
However, Celtic quickly reasserted themselves and looked equally as threatening as the Polish side, with midfielder Kris Commons coming close with an effort from 18 yards midway through the first half.
But the slackness prevalent in the Hoops defence would have been giving Deila palpitations and in the 36th minute, just after Teemu Pukki had missed a decent chance, the home side forged ahead all too easily.
Zyro's cross was headed down at the back post by Kucharczyk and, with the Celtic defence slow to react and Ambrose slipping, Radovic fired in from close range.
The Glasgow side were stunned but Legia's defence appeared as ropey as their own and a few minutes later McGregor drew a decent save from Kuciak with a sidefooted shot from the edge of the box.
However, when Ambrose was sent packing in the 44th minute for bringing down Kucharczyk just outside the box - the free-kick came to nothing - it looked an uphill task for the Hoops in the second half.
Striker Pukki made way for midfielder Beram Kayal and Commons moved up to lead the line, with captain Mulgrew dropping into defence.
There was no Legia onslaught but in the 54th minute Forster had to come to the rescue with a point-blank save from Zyro, who had been brilliantly set up by Radovic after getting the break of the ball inside the box.
Celtic were given a break when Vrdoljak sent his penalty wide of the target and slowly the Parkhead side began to sense that they could sneak another goal, although they continued to live dangerously at the back.
Commons had the ball in the Legia net but the offside flag had been up for an age
It was further evidence, though, that the visitors did not want to settle for the narrow defeat.
Emilio Izaguirre had replaced Berget and Leigh Griffiths came on for Commons but with six minutes remaining, Zyro headed in a cross from Radovic.
Forster 's penalty save from Vrdoljak was then negated when Kosecki made it 4-1 in the last minute of normal time to leave Celtic with it all to do next week.
Source : PA
Source: PA