Stephen Boyle, Auditor General for Scotland, said the pandemic had "significantly impacted" the programme.
The Scottish Government decided to delay the census last year as a result of the coronavirus crisis, moving it from March 2021 to March next year.
This has a "significant financial impact" on the National Records of Scotland (NRS), the public body responsible for planning and running the census.
Overall costs have increased by £21.6m (18 per cent), against the pre-pandemic estimate of £117m.
The Scottish Government has provided funding to cover this increase.
The census is one of the Government's biggest ICT projects.
There has been a census in Scotland every ten years since 1801, except 1941.
Audit Scotland said the NRS had acted to improve the management of the programme, but had faced "considerable challenges", including the impact of the pandemic on programme costs and difficulties recruiting suitably qualified staff.
Mr Boyle said: "The Covid-19 pandemic significantly impacted NRS’s plans for running Scotland’s census, resulting in ministers' decision to delay by a year.
"The census programme began to face challenges in 2018.
"NRS took steps to strengthen the programme and is now in a better position to deliver the census successfully in March 2022.
"It is crucial that NRS continues to monitor and manage ongoing risks to deliver the census, which is a key source of information to help the public sector plan services and funding."
Paul Lowe, registrar general and chief executive of the NRS, said: “Audit Scotland’s report highlights the significant risks that the pandemic posed to the delivery of Scotland’s Census in March 2021.
"NRS could not have compensated for a significant reduction in response rates due to the pandemic, or readily re-run the census if that occurred.
“In 2020 these risks prompted NRS to recommend to Scottish ministers to delay the census for a year, this was agreed.
“Additional costs were incurred as a result of the delay and it is right that the public are made aware of the full impact of the pandemic.
“The report confirms that NRS is taking appropriate action to manage the challenges associated with a programme of this scale and complexity and that we are in a strong position to deliver a successful census next year.”