On a night of comebacks in the Champions League, the defending champions left theirs far too late.
Chelsea is facing an uphill battle to qualify for the knockout stage after slumping to a 2-1 loss at Shakhtar Donetsk on Tuesday, the Premier League leaders' first defeat of the season in all competitions.
Manchester United also fell two goals down but staged a rousing fightback to win 3-2 at home to Braga and stay perfect after three group games. Barcelona has the same record after needing a goal deep into stoppage time by Jordi Alba to beat Celtic 2-1.
While those two European giants look assured of progress, Chelsea is battling to avoid becoming the first defending champion to fail to advance from its group.
"We need to win our two home games remaining for sure," Chelsea manager Roberto di Matteo said. "It just shows how the Champions League is a very competitive competition."
Shakhtar moved top of Group E with seven points, three above Chelsea, which was thankful that rival Juventus was held to a surprising 1-1 draw at Danish minnow FC Nordsjaelland. Juve has drawn all three matches to remain unbeaten this season at home and in Europe.
In its 20-year history, no team has retained the Champions League title.
Elsewhere, after its stunning 3-1 win over Bayern Munich last time out, unheralded Belarusian side BATE Borisov came back down to earth with a 3-0 home loss to Valencia to cede its lead of Group F to the Spanish team on goal difference.
Bayern, last season's beaten finalist, is also on six points in a tight group after Thomas Mueller's first-half penalty clinched a 1-0 victory at Lille.
Also, Spartak Moscow beat Benfica 2-1 to get off the mark in Barcelona's group and move a point behind Celtic in the battle for second. Cluj conceded a late equalizer to draw 1-1 at Galatarasary but is still second behind Man United in their group.
Barcelona won its 100th Champions League match and set a new club record of 18 home matches in Europe without a loss, but the win over Celtic was much tighter than expected.
With Lionel Messi in subdued form, Celtic took a shock 18th-minute lead when Javier Mascherano deflected a free kick into his own net and the Scottish side defended stoutly before Andres Iniesta equalized in the 45th for Barcelona.
