He spoke about how the “kindness and perseverance of the British spirit” had shown “how much we can achieve when we all pull together” during the coronavirus pandemic.
And with the UK vaccination programme having “roared into life in every corner of this country”, he said restrictions that have been in place across the UK for much of the last year could soon be removed.
Mr Johnson said: “In the coming months we will beat Covid – we will vaccinate everyone in our country and we will be able to remove restrictions.
“In the not-too-distant future, we will be able to reopen businesses, see friends in each other’s houses and hold our loved ones again.”
Mr Johnson, whose UK Government determines the restrictions for England only, was addressing the Scottish Conservative conference.
He told the virtual gathering that more than 1.8 million people in Scotland and 22.5 million people across the whole of the UK have now had at least their first dose of the vaccine.
The Prime Minister pledged: “We are resolved to keep that pace and accelerate it wherever possible, so that every single adult across the entire UK will have received their first injection by the end of July.”
His speech comes almost a year since the first lockdown, aimed at curbing the spread of the virus, was announced in March 2020.
And Mr Johnson said the “spirit” shown by the country over the last 12 months would “need to continue to show in the months and years ahead” as the country seeks to rebuild in the aftermath of the crisis.
But he said that the “team effort” shown in fighting Covid-19 had been “an effort that has shown the brilliance of every part of our union”.
Mr Johnson said that national effort has “brought our country together”, adding that “cutting-edge technology and science” in Scotland – where Valneva is working on a vaccine – was “helping to power the UK’s battle against coronavirus”.
Mr Johnson said: “This effort has shown what the UK can do: pulling together such a massive programme, the biggest in our peacetime history, over such a short period of time.”
He praised NHS staff for their work and also hailed the “organisational might” of UK armed forces, who he said had established 80 vaccination centres in Scotland and “been at the sharp end administering those vaccines in Edinburgh, Aberdeen and Glasgow”.
Mr Johnson said: “I want to use this opportunity to again say a massive thank you to all of the people that have been involved in delivering our vaccination programme.
“It shows that the great British spirit that saw us through so much adversary in the past, lives on in us today.”
While he accepted there had been “some differences of approach across the country” between his government and the devolved administrations, Mr Johnson said: “On the delivery of PPE, on testing and now the vaccination programme, we pulled together and worked as one United Kingdom.
“This demonstrated, quite simply, the United Kingdom’s collective strength.”
He stressed it was “work on behalf of the whole of the United Kingdom that secured our access to more than 400 million doses of seven vaccines, that are being distributed across our country”.