Massimo Donati's last-minute goal put Celtic on the brink of qualification from Group D of the Champions League after victory over Shakhtar Donetsk.
Celtic were always chasing the game after Brandao pounced on a John Kennedy mistake to fire the Ukrainian side ahead after just four minutes.
The Scots lost Lee Naylor and Kennedy to injury, but Jiri Jarosik blasted in an equaliser just before half time.
And Donati turned the ball home via a deflection in stoppage time.
It puts Celtic second in the Group D table ahead of Shakhtar, who must hope that they beat Benfica while Celtic lose away to AC Milan in the final games.
Artur Boruc and Paul Hartley had both survived late fitness tests to return to the Celtic side after missing Saturday's win over Aberdeen.
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Mark Brown and Donati dropped out, but the Italian midfielder was forced into the action after just 14 minutes following an injury to left-back Lee Naylor.
By then, Celtic were a goal behind following a nightmare start similar to when Shakhtar won the previous group game in Ukraine 2-0.
Kennedy mistimed his attempt at heading clear a long ball out of the Shakhtar defence, it struck Brandao and fell kindly for the striker to fire into the far corner of the net off the exposed Boruc's despairing hand.
That early success appeared to justify Shakhtar coach Mircea Lucescu's decision to go with two up front, matching Celtic manager Gordon Strachan's desire to go all out for a vital three points.
Shaktar were without Nery Castillo after a fall-out with Lucescu after the Mexican midfield playmaker defied orders not to take a penalty.
And, although the early goal visibly rocked Celtic's confidence, the home side began to take control of midfield and Scott McDonald fired just over from 20 yards, while a head flick from Jan Vennegoor of Hesselink drifted just wide.
But Shakhtar remained dangerous and Darijo Srna's 22-yard drive flew just a foot wide and Boruc relied on his instincts to turn a Brandao header from just six yards over the bar.
Celtic's chances appeared to take a further blow after 40 minutes, when Kennedy - not long back from serious injury - was stretchered off, with Steven Pressley making his first appearance of the season following back surgery.
However, a long ball out of defence from Gary Caldwell caused havoc in the Shakhtar defence and, when Razvan Rat and Dmytro Chygrynsky collided, Jarosik pounced to blast the ball into far corner.
The Czech midfielder ought to have grabbed himself a second goal in the closing seconds of the half but sent his diving header wide from five yards following a fine Aiden McGeady cross.
Celtic continued to press after the break, but chances were hard to come by and Jarosik was first to threaten with a glancing header just over the bar.
Just when it looked like it would end advantage to Shakhtar, Donati fired into the far corner from a McGeady cross to set Celtic Park alight.
Celtic: Boruc, Caldwell, Kennedy (Pressley 41), McManus, Naylor (Donati 16), McGeady, Scott Brown, Hartley, Jarosik, McDonald, Vennegoor of Hesselink (Killen 79).
Subs Not Used: Mark Brown, Zurawski, Sno, Caddis.
Booked: Jarosik, Vennegoor of Hesselink, Caldwell.
Goals: Jarosik 45, Donati 90.
Shakhtar Donetsk: Pyatov, Srna, Chigrinsky, Kucher, Rat, Ilsinho (Yezerskiy 83), Hubschman, Lewandowski, Jadson, Lucarelli (Gladkiy 88), Brandao.
Subs Not Used: Shust, Duljaj, Vukic, Shevchuk, Beylyk.
Booked: Brandao.
Goals: Brandao 4.
Att: 59,146
Ref: Bertrand Layec (France)
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