It was Georgios Samaras who got the crucial goal for Celtic shortly after the interval, although the Greek striker had a poor match.
Aberdeen will point to a couple of moments of bad luck. Scott Severin's shot in the fifth minute hit both posts before bouncing into Artur Boruc's arms and then, in stoppage time, Barry Nicholson chested the ball down to Alexander Diamond who shot home only for the referee to disallow the goal for handball by Nicholson.
It was a decision which Aberdeen protested vehemently against but in the end it was a nervous Celtic who secured all three points.
After Severin's early volley, Bobo Balde was just inches short of converting an Aiden McGeady cross. McGeady then had two shots but they were straight at the keeper.
Just before half-time, Samaras beat the offside trap but was too deliberate with his pass to Scott McDonald which was intercepted.
Samaras squandered an early chance in the second half, before Shunsuke Nakamura shot just wide.
McGeady went close with a shot which found the side-netting before Celtic went ahead when Barry Robson's inswinging corner was glanced home at the near post by Samaras who hardly had to get off the ground for his header.
Strangely Celtic started to sit back and substitute Paul Caddis did well to hook the ball clear from Lee Miller as the Aberdeen striker looked set to sweep home Nicholson's cross.
In a swift counter-attack McDonald's shot was palmed over by Jamie Langfield before McGeady headed over at the back from Samaras' cross.
Then came that controversial decision in stoppage-time which denied Aberdeen an equaliser.