He also formally invited people aged 60 and over to come forward for a jab.
Mr Hancock said the latest data proved the efficacy of both vaccines, urging: “When the call comes - get the jab.”
The health secretary said the number of new coronavirus cases is falling but the rate of decline has slowed, urging people to "keep sticking to the rules, let's not blow it now".
But the number of admissions to hospital was falling faster than that of cases - particularly among the older age groups who were vaccinated first.
"This is a sign that the vaccine is working," he said.
There was an even clearer sign in the data on deaths, Mr Hancock told the press conference, with the rate of decline in the older groups faster than in the under-80s.
"This shows, in the real world , across the UK right now that the vaccine is helping both to protect the NHS and to save lives."
"A single shot of either the Oxford/AstraZeneca vaccine or of the Pfizer vaccine works against severe infection among the over-70s with a more than 80 per cent reduction in hospitalisations", Mr Hancock said.
"In fact, the detailed data show that the protection that you get from catching Covid 35 days after a first jab is even slightly better for the Oxford jab than for Pfizer, albeit both results are clearly very strong."
The results "may also help to explain why the number of Covid admissions to intensive care units among people over 80 in the UK have dropped to single figures in the last couple of weeks".
The Health Secretary also defended the Government's border arrangements following the detection of the cases of the Manaus variant.
He said home quarantine measures were already in place and travel restrictions on Brazil had been imposed before the hotel policy was implemented.
Mr Hancock said: "All the evidence is that the five cases that we know about followed those quarantine rules and that, I hope, is very reassuring to people.
"There is no evidence that the sixth case did not follow those quarantine rules - we need to obviously get in contact with the person in question."
The test was taken on February 12 or 13 and "we haven't seen any further knock-on transmissions in the data".