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.
Reaction & as it happened
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.
Source – BBc