Health secretary Jeane Freeman initially gave a target of the end of January for one million doses, but delays to this have been attributed to supply issues.
Some 866,823 people had been given their first dose as of 8:30am on Monday. Scotland has also reached the milestone of 10,000 second doses of Pfizer vaccine administered.
Ms Sturgeon said she would like to see vaccine figures on Sundays improve, after this week showed a close to 50 per cent drop from Saturday to Sunday for the third week running.
Some 27,557 doses were given on Sunday, a significantly higher total than the week before, but almost half the number given on Saturday.
Last week doses dropped below 10,000 on Sunday, the lowest daily total on record.
Ms Sturgeon said then she was "absolutely clear” that any issues around Sunday figures must be resolved, as the aim for the programme was to continue at a “certain pace” every day of the week.
Ms Sturgeon this week celebrated the overall increase in daily figures, but said that she still hopes to see Sunday figures improve.
"We are working to ensure that boards on Sundays are picking up to their maximum capacity,” she said.
“I want to see that figure increase in the weeks ahead for some days, but I don't think we can take anything away from the significant acceleration of peace and progress that we've seen over the course of the week.”
More than 95 per cent of over-80s not living in care homes have been given a dose, following the Scottish Government’s deadline of Friday last week.
A total of 99.6 per cent of residents in older care homes have received a first jab, which is “a scale of uptake which none of us really believed would be possible”, said Ms Sturgeon.
Almost 67 per cent of people aged 75-79 and 29 per cent of people aged 70-75 have received a first jab.
The Scottish Government has faced renewed calls to pick up the pace of the vaccination programme.
Liberal Democrats leader Willie Rennie said: “It is good news that Scotland's rollout is slowly picking up pace, but the First Minister should cut the boasting about record daily figures when Scotland is still way behind England.
“Every day and every vaccine counts as the longer people are left unprotected, the greater the exposure to risk and the longer it will take to defeat the virus.
“Almost a million people should have been vaccinated by now, but we are way off that number because of the sluggish start to the rollout here.”
As of Monday morning, 10,519,729 people had been given a first dose of Covid-19 vaccine in England, along with 303,478 people in Northern Ireland and 603,976 people in Wales.
Throughout the vaccination rollout, Scotland has remained the nation with the lowest percentage of its adult population having been given a first dose, and this has not not yet changed, though the gap between Scotland’s progress and that of other nations has narrowed significantly.
So far just over 20 per cent of adults in Scotland have been given a first dose, compared to just under 22 per cent of those in Northern Ireland, around 23.5 per cent of those in Wales, and just under 24 per cent in England.
Ms Sturgeon has attributed the slower relative pace of the rollout in Scotland to the prioritisation of residents in care homes, which is a more time-consuming and logistically challenging vaccination exercise.
The programme “undoubtedly picked up pace considerably over the course of the last week”, Ms Sturgeon said on Monday.
"The pace of vaccination is very welcome good news and it will be one of the things that start to make a difference in the weeks to come,” she said.
"I know from many of the messages I've been getting that is providing comfort and relief not just to those who are being vaccinated, but to their friends, family, neighbours and loved ones as well.”
In response to a question about possible confusion due to health boards using a variety of methods to contact patients for their jag appointments, Ms Sturgeon said that by the end of this week all health boards should be operating with a centralised system to issue vaccination appointment invitations by letter.
However, GPs and health boards will continue to use “a number of different ways” to contact patients to get through the process as quickly as possible.
Ms Sturgeon urged the Scottish public to have faith in the AstraZeneca Covid-19 vaccine despite a new study from South Africa that found the jab had limited capacity to prevent mild cases of the virus.
Ms Sturgeon said the new study had been a “small trial” that predominantly had young, healthy adults as its subjects, adding: “People should continue to be vaccinated with AstraZeneca with confidence and that is a very strong message we want to communicate to the public.”
Earlier, Scotland’s national clinical director Jason Leitch said public confidence in the vaccine was “absolutely crucial”.
He told the BBC’s Good Morning Scotland that he was not “overly worried” by the new study from South Africa, and labelled its results a “data challenge” due to the young age of participants.