Officials previously announced that the Aberdeen Western Peripheral Route was targeted for a December opening, already a delay of over a year from its originally planned finishing date.
Northern parts of the road, which stretches for 36 miles, have already opened.
The project is being managed by Aberdeen Roads Limited, who are now targeting a January deadline.
Mr Matheson said: “This week, ARL informed us it was no longer able to meet its own deadline of concluding works at the Don Crossing before Christmas.
“This is just over a fortnight since it set this deadline publicly in Parliament.
“I have consistently urged caution and realism about ARL’s ambitious timescales.
“Unfortunately, as disappointing as this news is, it comes as no surprise to me it has been unable to achieve this.
“We have been here before with ARL.
“My officials and I will continue to work constructively and offer every assistance to get the remaining 4.5 miles open as soon as possible.
“I also want to pay tribute to the workforce who have undoubtedly been working hard during this most challenging time of year.”
He added: “More than 85 per cent of the AWPR is now open to traffic and is delivering immediate benefits which the people of the North East are clearly enjoying.
“I’m thrilled to be seeing story after story of people and businesses seeing substantial improvements to their journey times and reduced congestion across the city.”
Scottish Labour MSP Lewis Macdonald condemned the Scottish Government for announcing the delay to the AWPR after the Scottish Parliament closed for Christmas.
Mr Macdonald said: “Ministers admitted that they have known since Tuesday that the promise given to open the Don Crossing on the AWPR in December would not be met yet kept this further failure secret until the very end of First Minister’s Questions at Holyrood (on Thursday).
“This cynical ploy shows a profound disrespect for Parliament and a complete failure on the part of SNP ministers to be accountable to the people of Scotland.”