The Italian turned in a cross from Aiden McGeady deep into stoppage time as the Bhoys grabbed a win that had looked extremely unlikely for long spells.
An early goal by Brazilian Brandao stunned Celtic and first-half injuries to Lee Naylor and John Kennedy left Gordon Strachan's side in trouble.
Jiri Jarosik did manage an equaliser and, just when Shakhtar had settled for the draw, Donati struck.
Artur Boruc and Paul Hartley both returned to the Celtic starting line-up to give Strachan a pre-match boost.
But Celtic fell behind after four minutes when their defence was caught out by a long throughball. Kennedy headed the ball against Brandao who pounced to steer a shot beyond Boruc.
It was just the start that Celtic did not want and there was a further setback when Naylor went off after just 16 minutes with what looked like an ankle injury. Paul Hartley dropped back to take his place.
Some of Celtic's passing was slack as they were clearly unnerved by the early problems.
Scott McDonald tried to lift his team with a tremendous shot which dipped just over the crossbar after good work by McGeady.
Gary Caldwell then produced a perfect cross for Jan Vennegoor of Hesselink and his diving header slipped just past the post.
But Shakhtar were dangerous on the break and Darijo Srna cracked a shot narrowly wide with Boruc scrambling across his goal.
Boruc had to react swiftly to tip a Brandao header over his crossbar with the home defence under pressure from Jadson's corner.
Celtic then lost another defender before half-time with Kennedy stretchered off with what looked like a serious knee injury with Steven Pressley coming on to replace him.
But Celtic found a way back into the game when the Shakhtar defence were caught in two minds and Jarosik cashed in spectacularly as he crashed a left-foot shot high into the net.
Jarosik missed an easier chance in first-half stoppage time when he somehow headed wide from close-in.
It was cat-and-mouse in the second half with both teams desperate not to make another mistake and neither goalkeeper was particularly busy.
McGeady gave the ball away 30 yards from his own goal but Jadson's shot was a yard too high.
Boruc then did well to smother a low cross from Srna with Shakhtar looking menacing.
At the other end, Jarosik headed a Hartley corner over from just six yards out with 15 minutes left.
But just when Celtic looked to have run out of ideas, Donati came up with his dramatic winner.