Mr Johnson was addressing leaders including US president Joe Biden as he hosted a virtual summit of the group of rich industrialised countries.
He also called on the G7 nations – also including France, Germany, Italy, Japan and Canada – to join together in focusing on the need for action to stem global warming.
And he welcomed Mr Biden’s decision to recommit the US to the Paris Agreement on climate change, after his predecessor Donald Trump withdrew from the 2015 accord.
Today’s discussion was being held by video link because of the coronavirus pandemic, but Mr Johnson said he hoped to welcome the leaders to the UK in June for the full annual G7 summit in Carbis Bay, Cornwall.
Speaking after the UK gave first doses of Covid-19 vaccine to more than 16m people – more than in any other G7 state – Mr Johnson said: “Science is finally getting the upper hand on Covid, which is a great, great thing and long overdue.
“But there is no point in us vaccinating our individual populations – we’ve got to make sure the whole world is vaccinated because this is a global pandemic and it’s no use one country being far ahead of another, we’ve got to move together.
“So, one of the things that I know that colleagues will be wanting to do is to ensure that we distribute vaccines at cost around the world – make sure everybody gets the vaccines that they need so that the whole world can come through this pandemic together.”
Mr Johnson called on his fellow leaders to step up action on climate change at the UN Cop26 summit which the UK will host in Glasgow in November.
“I think that this is the right moment for us all to focus on the other great natural challenge about which we’ve been warned time and time and time again,” he said.
“We can’t ignore it – the warnings have been even clearer than they were for Covid – and that is the problem of climate change, and that’s why we’re going to be working very hard to get some great things done at G7 on our plans for the Cop26 summit that we are holding along with our Italian friends in Glasgow in November.
“It’s great, by the way, that Joe has brought the United States back into the Paris climate change accords – a great step forward.”