Celtic endured a frustrating night in Warsaw as a resolute Shakhtar Donetsk held them in a pulsating Champions League draw.
A deflected Rio Hatate effort had the dominant Scots deservedly ahead after just 10 minutes.
But a blistering run and finish from Mykhaylo Mudryk levelled the tie before the break.
Celtic spurned several chances to take the win in the second half with Giorgos Giakoumakis blazing over late on.
After losing 3-0 to Real Madrid in their opener last week in Glasgow, and drawing here, Celtic's tough start continues with a trip to Germany to face RB Leipzig in three weeks' time.
Celtic were hunting for their first Champions League win in almost five years. You have to go back to 27 September, 2017 for their last one - against Anderlecht in Belgium - and the path to victory was mapped out early for the Scottish champions.
On the eve of the game, both teams alluded to the fact Shakhtar, playing their group games in Warsaw due to the Russian invasion of Ukraine, would sit deep and look to strike on the break as they did against Leipzig last week in a surprise 4-1 win.
That approach was evident as the Scots poured forward with the energy and movement which gave Real Madrid an early fright last week, with the Ukrainians being suckered by their own gameplan after just 10 minutes.
A rapid counter saw a burst of orange jerseys spring forward into the Celtic half, only for Josip Juranovic's true clearance to zip long to Sead Haksabanovic. The Swede's control and pass to the in-rushing Hatate was perfect, with the Japanese darting into the box to squeeze a ball through the tightest of gaps off the sliding Artem Bandarenko and into the net.
Given Celtic had already gone close through Kyogo Furuhashi and Greg Taylor prior to the opener, they were in firm control as defender Mortiz Jenz's half volley from a knockdown was then parried away by Anatoliy Trubin.
But just as the Scottish Premiership leaders appeared in command, a blistering counter levelled the game. A neat ball inside Juranovic was latched on to by Mudryk. The strapping 21-year-old raced away from the chasing full-back, and crashed a left-foot shot high across a sprawling Joe Hart.
It could have been worse moments later as Marian Shved, once the forgotten man of Parkhead, slammed the ball low under Hart at the second attempt. Thankfully for his old team, Mudryk was flagged for offside before a cute cut back.
Celtic dictated most of the play in a frustrating second half. Shakhtar's flurries forward were much more tame, while their goal led a charmed life at times thanks to their visitors' wastefulness in front of goal and savvy keeping from Trubin.
Jota was denied soon after the break by the goalkeeper, substitute David Turnbull arced a shot just over, and Jota meandered his way through a glut of Shakhtar players before a deflected off-balance shot trundled out for a corner.
As the time ticked down, Celtic stepped it up again. Substitute Giakoumakis lashed wide when unmarked from 14 yards, and Daizen Maeda hid his head in the turf in frustration as he failed to turn in a drilled cross in the closing minutes.
And with only a few seconds left, Giakoumakis outjumped Trubin to head the ball goalward, only for a orange jersey to scramble it clear.
Man of the match - Greg Taylor
The Scot had to be on his toes to fend off any Shakhtar counters, and he drove forward with purpose all nightDominance needs to be converted into goals - analysis
It must not be forgotten in the aftermath of this draw that this Shakhtar team thumped RB Leipzig only last week. Not only that, but this is the Champions League against a side much more used to this stage in recent years.
However, the Celtic fans slipping into the Polish night here will do so with a feeling of frustration. Their team were dynamic, creative and, most importantly, in the lead. It was hard to see them getting suckered. But Shakhtar did what they do so well.
Let's be honest, this group is a three-way fight for second place, and Ange Postecoglou will be pressing upon his team the need to be more clinical. There were enough chances to have pulled away from the Ukrainians.
One point from two tough games is far from a disaster. But it's crucial Celtic convert their dominant periods in these games into goals if they are to realise aspirations of knockout Champions League football.
Match stats
After winning all four home games in the Champions League in 2017-18, Shakhtar have since won just one of their last 13 as the designated home side (D7 L5).Celtic have kept just one clean sheet in 34 away games in the Champions League - the most games for any side with only one clean sheet or fewer.They have a shot conversion rate of 4% in the Champions League this season, with that including an own goal. By contrast, they have a shot conversion rate of 20% in the Scottish Premiership this term.Aged 21 years and 252 days, Shakhtar winger Mykhailo Mudryk is the youngest Ukrainian to score in consecutive Champions League appearances since Andriy Shevchenko in October 1997 (21 years 2 days).Moritz Jens completed 109 passes (114 attempted) - the most on record for a Celtic player in a Champions League match (2003-04 onwards).Jota completed eight dribbles of nine attempted. This is the joint-most on record (2003-04 onwards) for a Celtic player in the Champions League along with Aiden McGeady v Benfica in October 2007 (8/12).What's next?
Celtic travel to Paisley on Sunday (12:00 BST) to take on St Mirren in a bid to keep up their 100% Scottish Premiership start. Their next Champions League match is away to Bundesliga side RB Leipzig (17:45) on 5 October.
Line-ups
Shakhtar Donetsk
Formation 4-1-4-1
81Trubin
23Maia Reis5Bondar22Matvienko26Konoplya
6Stepanenko
9Shved21Bondarenko8Sudakov10Mudryk
11Zubkov
81Trubin23Maia Reis5Bondar22Matvienko26Konoplya6Stepanenko9ShvedSubstituted forPetryakat 61'minutes21BondarenkoSubstituted forDurasekat 74'minutes8SudakovSubstituted forOcheretkoat 78'minutes10Mudryk11ZubkovSubstituted forTraoréat 61'minutesSubstitutes
1Shevchenko2Traoré4Kryvtsov14Sikan15Mykhaylichenko16Kryskiv17Durasek20Topalov27Ocheretko30Pyatov32Kozik34PetryakCeltic
Formation 4-2-3-1
1Hart
88Juranovic20Carter-Vickers6Jenz3Taylor
42McGregor41Hatate
9Haksabanovic33O'Riley17Neves Filipe
8Furuhashi
1Hart88Juranovic20Carter-Vickers6JenzBooked at 52mins3Taylor42McGregor41HatateSubstituted forTurnbullat 68'minutes9HaksabanovicSubstituted forMaedaat 45'minutes33O'RileySubstituted forMooyat 68'minutes17Neves FilipeSubstituted forAbadaat 86'minutes8FuruhashiSubstituted forGiakoumakisat 68'minutesSubstitutes
7Giakoumakis11Abada13Mooy14Turnbull25Bernabei28Abildgaard29Bain31Siegrist38Maeda49Forrest56Ralston57WelshLive Text
Match ends, Shakhtar Donetsk 1, Celtic 1.
Second Half ends, Shakhtar Donetsk 1, Celtic 1.
Attempt missed. Daizen Maeda (Celtic) left footed shot from the centre of the box is close, but misses to the left. Assisted by Josip Juranovic with a cross.
Substitution, Celtic. Liel Abada replaces Jota.
Attempt missed. Giorgos Giakoumakis (Celtic) left footed shot from the centre of the box is close, but misses to the left. Assisted by Greg Taylor.
Attempt blocked. Oleh Ocheretko (Shakhtar Donetsk) left footed shot from the right side of the box is blocked.
Corner, Celtic. Conceded by Valerii Bondar.
Attempt blocked. Daizen Maeda (Celtic) right footed shot from the centre of the box is blocked. Assisted by Giorgos Giakoumakis with a headed pass.
Foul by Cameron Carter-Vickers (Celtic).
Yukhym Konoplya (Shakhtar Donetsk) wins a free kick in the attacking half.
Substitution, Shakhtar Donetsk. Oleh Ocheretko replaces Heorhii Sudakov because of an injury.
Foul by Aaron Mooy (Celtic).
Heorhii Sudakov (Shakhtar Donetsk) wins a free kick in the defensive half.
Substitution, Shakhtar Donetsk. Neven Durasek replaces Artem Bondarenko.
Corner, Celtic. Conceded by Valerii Bondar.
Attempt blocked. Jota (Celtic) right footed shot from the centre of the box is blocked.
Daizen Maeda (Celtic) wins a free kick in the defensive half.
Foul by Mykhailo Mudryk (Shakhtar Donetsk).
Attempt missed. David Turnbull (Celtic) right footed shot from outside the box is high and wide to the left. Assisted by Jota.
Substitution, Celtic. Aaron Mooy replaces Matt O'Riley.