Some shambolic defending was the blame for the Hoops succumbing to their technically superior opponents and the Parkhead club can count themselves lucky that the 2-0 scoreline was able to somewhat mask the fact that they were definitively outclassed.
Mircea Lucescu had the luxury of £40million spending spree in the summer to help bolster his side's chances in the competition and astonishingly the Donetsk coach was able to leave £16million worth of talent in the Mexican striker Nery Alberto Castillo on the bench.
Celtic manager Gordon Strachan preferred to leave Aiden McGeady in reserve whilst pairing Chris Killen and Scott McDonald up front in their debuts in the group stages of the competition.
Five minutes into the match and some woeful defending from the Scottish champions gifted the lead to their Ukrainian opponents.
Brandao was the grateful recipient of Scottish charity as a short back pass from Scott Brown to Stephen McManus allowed the striker to nip in and slip the ball past an ill-prepared Artur Boruc.
Cristiano Lucarelli sensationally doubled the Ukrainian club's lead two minutes later as the Hoops defence were still trying to regroup.
Lee Naylor struggled to close down Razvan Rat allowing the full back the time to cross into the box. The Italian striker managed to get in front of his marker to head powerfully into the net from six yards out.
Rat continued to torment Celtic down the left, again flighting the ball into the box for Lucarelli but the 32-year-old striker's effort was parried easily by Boruc.
Brandao once more had the ball in the net in the 19th minute but the linesman incorrectly ruled that the Brazilian was offside.
Celtic's first real foray into opposition territory ended when McDonald picked up Massimo Donati's pass before trying his luck from the edge of the box, with the little Aussie's strike flying just wide of the upright.
Celtic switched things around tactically in midfield in an effort to stifle Shakhtar's attacking forays with Paul Hartley occupying the same holding role that he performed superbly for Scotland in Paris last week.
Gordon Strachan must have thought his men had pulled a goal back in the 70th minute as Jan Venegoor of Hesselink rose to meet a Naylor cross, but the Dutchman's header flashed agonisingly wide of the upright.
Gary Caldwell had to produce a last-ditch goal-line tackle with nine minutes remaining to keep the score line respectable as the defender nipped in just ahead of Brandao, after Boruc had parried an effort from Fernandinho.
Substitute Chris Killen almost managed to score his first ever goal in the Champions League, but the New Zealander could only watch in disbelief as Andriy Pyatov managed to stick out a leg to divert the striker's effort.