Constitution secretary Angus Robertson said 604,000 Scottish households had yet to submit a response.
He confirmed the National Records of Scotland (NRS) will now accept returns until the end of May – an extension of four weeks – at an extra cost of almost £10 million.
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Mr Robertson stressed filling out the form was a legal responsibility, adding: “Failure to meet this responsibility can result in prosecution, which could lead to a criminal record and a fine.”
Scottish Conservative MSP Donald Cameron said the SNP’s handling of the census “has been nothing short of disastrous”.
Speaking in Holyrood, he said: “Needlessly delayed by a year, costing taxpayers an extra £30m and as of today around a quarter of households in Scotland still to respond.
"The SNP ignored our calls to hold the census in sync with the rest of the UK, when they would have benefited from the UK-wide publicity campaign – a campaign which managed a 97 per cent response rate.”
He added: “This all represents a significant failure by the Scottish Government, which will have serious, long-term implications for public policymaking.”
The census was delayed in Scotland because of the pandemic, but went ahead in England and Wales with high levels of completion.
There are concerns that decoupling the survey from other parts of the UK has reduced awareness.
It is also the first online census, although paper copies can be requested by those who do not have access to the internet.
Mr Robertson said he did not believe there would be further delays.
Earlier, he told MSPs 77.2 per cent of Scottish households have provided a census return, which he called “a substantial figure with everything happening in the world right now”.
He said: “I understand that many people may be dealing with other concerns.
"Recent world events have caused anxiety for many and have remained a focus for the media, quite rightly, in recent weeks.
"Closer to home people are still dealing with the impacts of Covid-19 and the cost-of-living crisis and given these challenges I appreciate another ask of people is difficult.”
However, he said “a census requires a high response rate and one that captures the diversity of our communities and their needs appropriately”.
Mr Robertson said “a significant multi-channel awareness campaign” was launched in February, including social media, radio and TV advertisements.
He said census extensions had taken place elsewhere in the world, including in Poland, Japan and the US, while Northern Ireland “continued to accept returns after the deadline”.
The constitution secretary added: “An extension to the census collection period remains a legitimate, and often used, process to facilitate engagement with the census process.”
NRS chief executive Paul Lowe said: “Census data is vital in informing decisions about services that affect us all. We have put in place a number of additional interventions to support those who have yet to complete a return.
"This includes a range of additional household reminders. Our field team have already undertaken more than 750,000 household visits to support those who have not completed, and are continuing to make these visits.”