Health secretary Jeane Freeman issued a statement to thank all those involved in the rollout of the programme as it reached the milestone on Wednesday.
The target of one million vaccines was initially set by Ms Freeman for the end of January, but was delayed due to supply issues.
Ms Freeman warned a UK-wide drop in supply towards the end of February would cause the rollout to slow, but said the programme was delivering “ahead of expectations”.
She said: “Our aim is to vaccinate as many people as possible with both their first and second doses as quickly as possible, but the speed at which we can do this depends on supply and we expect a dip in supplies UK-wide towards the end of this month.
“We hope to see a significant drop in the disease due to the vaccination programme. However, this will take a number of months to evaluate fully. In the shorter term, we are monitoring the uptake rate, but we also have a comprehensive surveillance system in place to monitor outcome of vaccine efficacy and disease reduction.”
John Loch, 69, was among those vaccinated at the NHS Louisa Jordan in Glasgow as the milestone was reached.
“I am delighted to have received my vaccine, particularly on the day Scotland reached a million doses,” he said. “It is exciting to be a part of this milestone moment.
“I would like to thank all the staff at the Louisa Jordan for being so reassuring and making the process so simple and straightforward.”
The total number of people given a first dose of vaccine dropped from Monday to Tuesday after some appointments were cancelled due to the weather. But Ms Sturgeon said it had been an “extraordinary” feat to complete 57,447 vaccinations despite the conditions.
NHS Lothian said some vaccinations of people at home had been rescheduled due to the snow, and a number of vaccination centres in NHS Fife and NHS Tayside were temporarily closed on Tuesday evening.
Asked how far people should be expected to travel to get to a vaccination appointment at First Minister’s Questions on Wednesday, Ms Sturgeon cited around 35 miles.
"Every effort has been made and will continue to be made to minimise travel times and distances to vaccination centres where that is possible,” she said.
"I know that some residents in areas such as East Lothian have had to travel to central Edinburgh locations and that, for people in some parts of East Lothian, that might be a distance of around 35 miles.”
Ms Sturgeon said those offered an appointment that was “unsuitable” for them may be able to reschedule.
Iain Gray, Labour MSP for East Lothian, said some of his constituents from Dunbar or North Berwick had been asked to travel past two vaccination hubs in East Lothian to get to the Edinburgh International Conference Centre (EICC) or Royal Highland Showground (RHS) – a round trip of 40 miles involving two or three bus journeys or a return taxi fare of £120.
He said: “When people phone the helpline, they are routinely and repeatedly told that nothing can be done and that no closer appointments are available. Meanwhile, they hear stories of Midlothian residents being sent to Haddington in East Lothian for their vaccination. We have the whole roll-out of second doses still to come. Will the First Minister intervene and sort this out?”
Ms Sturgeon said the Scottish Government was trying to achieve a “balance” between speed and accessibility of the programme.
"The arrangements will never be perfect for people, because we are trying to vaccinate the entire adult population as quickly as possible,” she said.
"Most of the people who contact me recognise that, but equally we recognise that we need to make sure that people are not being asked to travel inordinate distances or being put in a position in which it is genuinely impractical for them to attend a vaccination appointment."
One Gullane resident who was given a vaccination appointment at the EICC, 20 miles away, said she had been “surprised at having to travel so far”.
"I entirely accept that there is a need for mass vaccination centres, but friends in East Lothian of the same age group are being vaccinated at East Lothian hospital,” she said.
NHS Lothian’s first drive-through vaccination centre opened at Queen Margaret University (QMU) in Musselburgh on Wednesday.
The centre has capacity to vaccinate 720 people a day.
Pat Wynne, director of nursing for primary and community care at NHS Lothian, said the opening of the drive-through centre was a “big day” for both NHS Lothian and East Lothian.
"It is a unique site for us and it will allow us to deliver more vaccines to the people we serve,” she said.
QMU is the fourth mass vaccination centre to open in Lothian, following the EICC, Strathbrock Partnership Centre and the Royal Highland Centre.
The First Minister also faced questions on vaccine wastage after staff at Whitehill community centre in Hamilton had to discard up to 98 doses of Pfizer vaccine last week due to appointments not being filled.
"I am never going to stand here and say that, in a programme of such a scale, everything every single day is perfect and there are no glitches or things that go wrong,” Ms Sturgeon said.
"That is not going to be the case. The exercise is the biggest peacetime logistical exercise that we have ever undertaken in Scotland. The same is true in the other UK nations.
"When things go wrong, as happened in Fife this week, we have to take action quickly to resolve that.”