The 9ft bronze figure was hoisted onto a lorry to be taken to the Edinburgh-based firm Powderhall Bronze, where it will be cleaned and repaired.
When it is returned in about five weeks it will be put on a raised plinth in a bid to prevent further damage.
It is hoped that CCTV cameras installed nearby will also help deter vandals.
The £30,000 statue, which usually stands outside the Buchanan Galleries shopping centre, has been defaced on several occasions.
Council officials have now decided to have it fully repaired at a cost of £25,000.
Masonry contractors CBC Stone, who removed the statue, are donating £5,000 towards the cost.
'Graffiti removed'
CBC Stone's Graeme Frame said the next stage of the operation would be carried out by Powderhall Bronze, a firm with specialist expertise in the conservation of outdoor statues.
"He's going back there really just to get the spectacles put back on and to get the graffiti removed, and repatinated," he said.
"That should take three or four weeks, during which time we will be taking the existing base away."
A new two-metre-high plinth will then be erected in the same position.
Joseph Mullin, who watched the operation, said he was looking forward to seeing Mr Dewar's statue back in place on a bigger plinth.
"I'm sorry to see him on his back like this. You almost feel he should have been put in a box," he said.
Other people watching expressed concern about whether a bigger plinth would deter vandalism.
One woman said the late first minister, who was dubbed the 'father of devolution' deserved more respect.
"Hopefully they will leave it alone. Because he was a man who has done a lot for Scotland and to damage him at all is an absolute disgrace," she said.
The statue of Mr Dewar, who died of a brain haemorrhage in 2000, was officially unveiled in May 2002.
Within days Mr Dewar's glasses were left twisted and bent by vandals and in a more recent incident his leg was spraypainted.
^ Back to top | BBC Sport Home | BBC Homepage | Contact us | Help | ©
~RS~q~RS~~RS~z~RS~53~RS~)