Ms Sturgeon added that is is hoped all Scots over 80 will be offered a first dose in the next four weeks.
Vaccination of the over 80s in the community began in Scotland on Thursday, with the first doses given at Pentlands Medical Centre in Edinburgh.
Just under 21,500 people in Scotland were given a Covid-19 vaccine in the fourth week of the programme’s rollout ending on January 3, bringing the total number of vaccinations to 113,500.
Vaccination numbers dropped significantly in the fourth week, which coincided with New Year and a sudden change to the dosing regime of the Pfizer vaccine announced on Hogmanay.
The figures do not take into account the AstraZeneca vaccine, delivery of which began on January 4.
Just over 18,500 people were given a dose in the first week, with rollout stepped up to around 38,000 doses delivered in the second week and 35,500 in the third week.
Ms Sturgeon urged those due to receive a vaccine not to contact their GP practice, but to wait until they are contacted.
She added that a total of 1,100 vaccination sites will soon be set up around Scotland, most of which will be GP surgeries.
The Scottish Government now has two Covid-19 vaccines available, with the first doses of the AstraZeneca jab given on Monday January 4, just days after it was approved for use in the UK by the Medicines and Healthcare Products Regulatory Agency (MHRA).
First Minister Nicola Sturgeon has said Scotland faces a “race” between rising Covid-19 cases and the rollout of a vaccine.
It is hoped the country will have access to 900,000 doses by the end of January, split roughly equally between the two vaccine types.
Ms Sturgeon said the Scottish Government “does not yet have certainty” on supply schedules beyond January, but hopes that by early May everyone on the first vaccine priority list according to the Joint Committee on Vaccination and Immunisation (JCVI) will have been offered a vaccine.
This timeline will require a significant increase in the number of doses delivered a week, as at the current rate a first dose will not be given to all 2.7 million people on that list until mid 2022, without taking into account second doses.
Second doses of the Oxford AstraZeneca vaccine will be given 12 weeks after the first.
Those for the Pfizer vaccine were originally planned to be given three to four weeks after the first, but will now be administered after up to 12 weeks.
This change, which has been recommended by the JCVI, has been accepted by the Scottish Government despite some criticism over disruption to patients and vaccinators.
Asked about the change in the Coronavirus update on Thursday, Ms Sturgeon said: “These are decisions which are not taken lightly, but right now we’re in a race against the virus and we’ve got to get as many people as much immunity to it as possible.”