The prime minister said the UK has ordered more than 350 million doses of various coronavirus vaccines, three of which have been found to be "highly effective".
If those vaccines are given medical approval and can be mass produced, then, "with a favourable wind" the NHS "should be able to inoculate" the "vast majority" of the people who need the most protection by Easter.
"That would make a very substantial change to where we are at the moment", he said, but added: "I don't want to give anymore hostages to fortune than that."
He made the remarks when asked by ITV News Political Editor Robert Peston how long it could be before all coronavirus restrictions are removed.
The prime minister hailed the development of vaccines by Oxford University, Pfizer/BioNTech and Moderna, but urged people to continue following rules and "fight down any over-optimism".
"We can hear the drumming hooves of the cavalry coming over the brow of the hill but they are not here yet," he said.
"Even if all three vaccines are approved, even if the production timetables are met and vaccines notoriously fall behind in their production timetables it will be months before we can be sure that we have inoculated everyone that needs a vaccine."
He described the rollout of any vaccine as a "big logistical challenge", adding: "That's why I wanted to enter this important note of caution.
"We haven't got these things yet, even when we do get them it will take a long time, it will take a while, before we can get the shots in the arms, where they're needed."
What you need to know about each vaccine:
The University of Oxford and AstraZenica vaccine is up to 90% effective, can be stored at fridge temperature and is the cheapest, costing just £4 a dose. The UK government has ordered 100 million doses, with four million ready to go.
The Pfizer/BioNTech vaccine has up to 95% efficacy when it comes to immunisation, requires to be stored at -70C degrees and will cost £15 a dose. The UK has ordered 40 million doses, 10 million of which will be available by the end of the the year.
The Moderna vaccine is up to 94.5% effective, requires to be kept at -4C and will cost between £24 and £28 per dose. The UK has five million doses on order.
All three vaccines require two separate inoculations to be effective.
Oxford Vaccine Group director Professor Andrew Pollard said it has been "a very exciting day" after trials showed the vaccine, which the UK has ordered 100 million doses of, could be up to 90% effective in preventing people from catching coronavirus.
Mr Johnson held the press conference hours after a speech in the House of Commons, in which he updated MPs on his plan for England to enter a tightened system of tiered coronavirus restrictions when lockdown ends on December 2.
In all areas of England non-essential shops, gyms, hairdressers, beauty parlours, and the wider leisure sector will be able to reopen, and collective worship, weddings and outdoor sports can resume.
What can you do in each tier from December 2? The new rules in England at a glance:
Tier 1: Up to six people can meet indoors or outdoors. Pubs and restaurants can open, with last orders at 10pm and closing at 11pm.
Tier 2: No mixing indoors, apart from support bubbles. Up to six people can meet outdoors. Pubs and restaurants can open, with last orders at 10pm and closing at 11pm - but alcohol can only be served with a substantial meal.
Tier 3: No mixing indoors. People can only meet outdoors in limited places such as parks and public gardens. Pubs and restaurants must close, with the exception of takeaway sales.
In all tiers, non-essential shops, gyms, hairdressers and personal care services can open. People in all tiers to work from home where they can do so. Full details on what you can do in each tier can be found here.
Mr Johnson said the tier system was being strengthened because restrictions before the lockdown were "not quite enough" to reduce the reproduction rate of coronavirus (the R number) below one.
"More regions will fall at least temporarily into higher levels than before", the prime minister said, and the rules under each tier will be non-negotiable.