Councillors met yesterday to discuss the issue and 93 voted for the motion to delay the plans, with 7 abstaining and none voting against.
The U-turn comes shortly after from his position, due to ongoing internal disputes over the plans.
Ex-deputy leader Jim Currie, who had briefly resigned over the outsourcing strategy, was appointed as the council's new leader last week.
Currie had sent a damning resignation letter to Robertson, claiming that he had 'pushed the cause of retaining Council control over joint ventures as far as [he could] with the Cabinet' and that the financial risks involved with the 'rush' into strategic partnerships was 'unacceptable'.
Outsourcing of the council's services was due to go to either BT or CSC.
Currie implied that Robertson was unwilling to back down on part-privatising the Council whilst he was in power. He said: "I know you will never let go. I could not leave local government with billions of pounds of Cornish taxpayers' money at risk and on my conscience."
The Council has now said that Currie needs to investigate fully, as a matter of urgency, all reasonable alternative methods of delivering the services covered by the proposals.
It said that these plans need to address the need to make efficiency savings and to generate income.
Under the previous proposals, BT or CSC would have been legally required to deliver at least £5 million of savings over the next two years and to create a minimum of 500 net new jobs in Cornwall.