The Dons made three changes to the side that drew with Hearts last Sunday, with Steve Lovell being replaced up front by veteran Craig Brewster, while Andrew Considine came in for the injured Lee Mair.
Bhoys boss Gordon Strachan also made changes following his side's 1-1 draw on Wednesday, with Artur Boruc returning in goal while Aiden McGeady came in from Paul Hartley. The other change came at the back with Gary Caldwell in for John Kennedy.
The home side had a penalty claim turned down early on, following Chris Clark's burst into the box, he seemed to be impeded by Lee Naylor but referee Charlie Richmond waived the claims away.
A minute later Aberdeen won a free-kick 20 yards from goal and Jamie Smith curled the ball towards the top corner but Boruc got across his goal to make a great one-handed save.
On six minutes Scott McDonald found Jan Vennegoor of Hesselink ten yards from goal but he dragged his shot wide of the target.
The first goal of the game came after 24 minutes when Michael Hart sent Barry Nicholson clear down the right-hand side. His low cross was met at the far post by Brewster, who slotted the ball home from four yards.
Right on the stroke of half-time Aberdeen almost doubled their advantage following a mistake by Boruc, who allowed Darren Mackie's 20-yard shot to squirm from his grasp and head goalwards, but the big keeper managed to recover and gather the ball on the line.
At the start of the second half Mackie had another chance to put the Dons two ahead, but his low shot flew into the sidenetting.
On 57 minutes Celtic's Shunsuke Nakamura flashed a trademark free-kick back off the underside of the bar as the Bhoys pressed forward.
Celtic did square things on 62 minutes when Aiden McGeady found Massimo Donati inside the box and he swivelled to fire home from 18 yards.
Five minutes from time Nakamura played a pass to Donati who in turn played it on to substitute Miller and his 12-yard low drive on the angle beat Derek Soutar and put the visitors ahead for the first time.
Then in the second minute of injury time Miller picked the ball up fully 30 yards from goal before curling a low shot beyond the outstretched arms of Soutar to secure the points for the champions.